J. Ma h. Anal. Appl. 514 (2022) 126335
Con en s lis s a ailable a ScienceDi ec
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www.else ie .com/loca e/jmaa
Regula A icles
Ex eme poin s o Lo enz and ROC cu es wi h applica ions o
inequali y analysis
Ampa o Baíllo a, Ja ie Cá camo b,∗, Ca los Mo a-Co al a
aDepa amen o de Ma emá icas, Uni e sidad Au ónoma de Mad id, 28049 Mad id, Spain
bDepa amen o de Ma emá icas, Uni e sidad del País Vasco, Ap do. 644, 48080 Bilbao, Spain
a i c l e i n o a b s a c
A icle his o y:
Recei ed 12 No embe 2021
A ailable online 16 May 2022
Submi ed by S. Geiss
Keywo ds:
Ex eme poin s
Gini index
Lo enz cu e
Lo enz o de ing
Inequali y
ROC cu e
We find he ex eme poin s o he se o con ex unc ions :[0, 1] →[0, 1] wi h a
fixed a ea and (0) =0, (1) =1. This collec ion is o med by Lo enz cu es wi h a
gi en alue o hei Gini index. The analogous se o conca e unc ions can be iewed
as Recei e Ope a ing Cha ac e is ic (ROC) cu es. These unc ions a e ex ensi ely
used in economics (inequali y and isk analysis) and machine lea ning (e alua ion
o he pe o mance o bina y classifie s). We also compu e he maximal L1-dis ance
be ween wo Lo enz (o ROC) cu es wi h specified Gini coefficien s. This esul
allows us o in oduce a bidimensional index o compa e wo o such cu es, in a mo e
in o ma i e and insigh ul manne han wi h he usual unidimensional measu es
conside ed in he li e a u e (Gini index o a ea unde he ROC cu e). The analysis
o eal income mic oda a illus a es he p ac ical use o his p oposed index in
s a is ical in e ence.
© 2022 The Au ho s. Published by Else ie Inc. This is an open access a icle
unde he CC BY-NC-ND license
(h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/).
1. In oduc ion
Gi en a non-emp y, compac and con ex se o a locally con ex space, he collec ion o i s ex eme poin s
plays a p ominen ole in op imiza ion and ma hema ical p og amming. By Baue ’s maximum p inciple
(see, e.g., Phelps [25, P oposi ion 16.6] o Alip an is and Bo de [1, 7.69]) any con ex, uppe -semicon inuous
unc ional defined on his se a ains i s maximum a an ex eme poin . In o he wo ds, we can sea ch o
maximize s o such unc ionals wi hin he se o ex eme poin s. The ele ance o ex eme poin s can be
also comp ehended h ough he K ein–Milman heo em (see, e.g., Simon [30, Theo em 8.14]), which is a
cen al esul in con ex analysis. This heo em affi ms ha he o iginal se is ac ually he closed con ex
hull o i s ex eme poin s. The e o e, we can e ie e he en i e con ex se by knowing he (usually much
smalle ) subse o ex eme poin s. The p ac ical applica ion o hese powe ul esul s goes h ough he
*Co esponding au ho .
E-mail add ess: ja ie .ca [email protected] (J. Cá camo).
h ps://doi.o g/10.1016/j.jmaa.2022.126335
0022-247X/© 2022 The Au ho s. Published by Else ie Inc. This is an open access a icle unde he CC BY-NC-ND license
(h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/).
2A. Baíllo e al. / J. Ma h. Anal. Appl. 514 (2022) 126335
explici compu a ion o he ex eme poin s o he con ex se unde s udy, which is usually a difficul ask
in infini e-dimensional spaces.
As a ma e o ac , one o he aims o his wo k is o find he ex emal poin s o some infini e-dimensional
collec ions o unc ions. Specifically, we define
L={:[0,1] →[0,1] : con ex and (0) = 0,(1) = 1},(1)
R={ :[0,1] →[0,1] : conca e and (0) = 0, (1) = 1},(2)
and we conside
La={∈L:=(1−a)/2},a∈[0,1],(3)
Ra={ ∈R: =a},a∈[1/2,1],(4)
whe e ·is he usual no m in L1=L1([0, 1]). We will p o ide a de ailed desc ip ion o he main p ope ies
o Laand Ra, which a e ac ually compac and con ex se s in he space L1; see P oposi ion 1.
Ou main mo i a ion lies in he ac ha he cu es in Laand Raappea epea edly in many applied
disciplines. On he one hand, an impo an in e p e a ion o he unc ions in Lis as Lo enz cu es o posi i e
and in eg able andom a iables. In addi ion, Lais he collec ion o Lo enz cu es wi h Gini index a. In
economics, Lo enz cu es a e ex ensi ely used o p o ide a g aphical ep esen a ion o he dis ibu ions o
income o weal h in popula ions, while he Gini index is pe haps he mos p ominen inequali y measu e. On
he o he hand, Rcan be iewed as he se o Recei e Ope a ing Cha ac e is ic (ROC) cu es. ROC cu es
a e employed o e alua e he quali y o classifie s in p obabili y o ecas s (see o example Fawce [15]) and
appea in many scien ific disciplines in which classifica ion o bina y ou comes is ele an (medical diagnos ic,
c edi sco ing, classifica ion o financial ansac ions, and so on). The Gini coefficien o a ROC cu e is
defined as wice i s a ea minus 1. Howe e , he a ea unde he ROC cu e (AUC sco ing) is used mo e
equen ly as a measu e o he global accu acy o he unde lying classifie ; see Sec ion 2 o de ails.
A second goal o his wo k is o quan i y how “ a ” wo o he cu es in (1)(o (2)) can be om
one ano he . Specifically, gi en wo Lo enz (o ROC) cu es wi h fixed Gini indices, we wan o compu e
he maximal L1-dis ance be ween hem. In o he wo ds, o a, b ∈[0, 1], we a e in e es ed in compu ing
he dis ance be ween he se s Laand Lb(o Raand Rb, o a, b ∈[0, 1/2]). This ques ion u ns ou o
be an infini e-dimensional con ex maximiza ion p oblem wi h wo linea cons ain s. We will sol e his
op imiza ion p oblem by a ca e ul analysis o he dis ance be ween he ex eme poin s o he conside ed
se s. This app oach also allows us o iden i y he maximize s o he unde lying unc ional.
This pape is s uc u ed as ollows: Sec ion 2 e iews he main in e p e a ions o he elemen s in Laand
Ra. We ecall he defini ion o he Lo enz cu e and he Gini index o an in eg able a iable, as well as
he main elemen s o desc ibe ROC cu es. In Sec ion 3, we de e mine he se o ex eme poin s o Laand
Ra. Sec ion 4is de o ed o he compu a ion o he maximal L1-dis ance be ween wo o hese se s. We
also iden i y ex emal cu es, ha is, unc ions o which his maximal dis ance is a ained, and show hei
connec ion wi h s ochas ic o de ings. As an applica ion, in Sec ion 5we in oduce a bidimensional inequali y
index o compa e diffe en cha ac e is ics o wo Lo enz o ROC cu es and enume a e i s main p ope ies. In
he case o Lo enz cu es, his new index measu es simul aneously inequali y and dissimila i y, while o ROC
cu es, i indica es accu acy and dissimila i y as well. We conside he empi ical e sion o his inequali y
index o be used in p ac ice. We also p opose o combine a boo s ap app oach wi h a non-pa ame ic se
es ima ion echnique o ob ain a confidence egion o he popula ion index. The p ocedu e can be u he
applied o es simple hypo heses ela ed o he index. These echniques ha e been implemen ed in he
so wa e R and a e illus a ed ia he analysis o some eal income mic oda a samples om Spain. Finally,
he p oo s o he main esul s a e collec ed in Sec ion 6.
A. Baíllo e al. / J. Ma h. Anal. Appl. 514 (2022) 126335 3
2. Lo enz and ROC cu es and he Gini index
In his sec ion we ecall he defini ions o Lo enz and ROC cu es and he Gini index, as well as hei
connec ions wi h he se s La(in (3)) and Ra(in (4)). Also, we b iefly desc ibe he scien ific fields in which
hese concep s a e widely used.
2.1. Lo enz cu es and he Gini index in economics
Le Xbe a posi i e andom a iable wi h fini e mean μ >0and cumula i e dis ibu ion unc ion
F(x) =P(X≤x), o x ≥0. The Lo enz cu e o he a iable X(o o he dis ibu ion F) is
( )= 1
μ
0
F−1(x)dx, 0≤ ≤1,(5)
whe e
F−1(x)=in {y≥0:F(y)≥x}(6)
(0 <x <1) is he quan ile unc ion o X, ha is, he gene alized in e se o F.
I Xmeasu es income in a popula ion, o each alue ∈[0, 1], he unc ion in (5)gi es us he (no malized)
o al income accumula ed by he p opo ion o he poo es in ha popula ion. No e ha F−1is non-
dec easing, μ =1
0F−1(x) dxand ( ) =F−1( )/μ a.e. ∈(0, 1). In pa icula , is con inuous excep
pe haps a he poin 1and has posi i e second de i a i e a.e. Mo eo e , as he quan ile unc ion in (6)
cha ac e izes he p obabili y dis ibu ion, de e mines he dis ibu ion o he unde lying a iable up o a
(posi i e) scale ans o ma ion. Explici analy ic exp essions o he Lo enz cu es o he usual pa ame ic
dis ibu ions can be ound in Kleibe and Ko z [20, Sec ion 2.1.2].
I is easy o check ha he se Lin (1)is he closu e (wi h espec o he poin wise con e gence) o
he se o Lo enz cu es o posi i e and in eg able andom a iables wi h s ic ly posi i e expec a ion. Fo
simplici y, we will e e o Las he class o Lo enz cu es. By con exi y, o e e y ∈Li holds ha
pi ≤≤pe,(7)
whe e
pe( )= (0 ≤ ≤1) and pi( )=0,i 0 ≤ <1,
1,i =1.(8)
Fig. 1shows a g aphical ep esen a ion o he inequali ies in (7). The unc ion pe is called he pe ec
equali y cu e as i co esponds o he Lo enz cu e o a Di ac del a measu e, i.e., he p obabili y measu e
co esponding o a popula ion in which all indi iduals ha e equal (and posi i e) incomes. Addi ionally, pi
is he pe ec inequali y cu e because i can be iewed as he limi (when he o al numbe o indi iduals
ends o infini y) o Lo enz cu es in fini e popula ions whe e only one pe son accumula es all he weal h.
No e ha he unc ion pi defined in (8)(see also Fig. 1), which is no a p ope Lo enz cu e, belongs o L.
In p ac ice, i is e y common o syn hesize he in o ma ion o he Lo enz cu e in a single nume ical
alue ha quan ifies income inequali y. Diffe en cha ac e is ics, unc ionals and alues o he Lo enz cu e
a e employed o cons uc hose inequali y indices; see A nold and Sa abia [4]. The mos popula inequali y
measu e de i ed om he Lo enz cu e is he Gini index. This index has a as numbe o in e es ing
4A. Baíllo e al. / J. Ma h. Anal. Appl. 514 (2022) 126335
Fig. 1. A Lo enz cu e oge he wi h he pe ec equali y and inequali y cu es.
in e p e a ions and ep esen a ions; see Yi zhaki and Schech man [35, Chap e 2]. One possible way o
define i is he ollowing:
G()=2
1
0
( −( )) d =1−2.(9)
The e o e, we ha e ha o a ∈[0, 1], he se Ladefined in (3)is p ecisely
La={∈L:G()=a},(10)
he collec ion o Lo enz cu es wi h Gini index a.
Obse e ha
G()= −pe
pe −pi.(11)
The denomina o in (11)equals 1/2 ( he maximum L1-dis ance be ween Lo enz cu es) and ac s as a
no malizing cons an so ha 0 ≤G(X) ≤1. G aphically, G()is wice he shaded a ea in Fig. 1.
The Gini index has many con enien p ope ies: i is scale- ee (because he Lo enz cu e is i sel in a ian
unde posi i e scaling); i can be compu ed whene e he conside ed andom a iable is in eg able (so fini e
second momen is no necessa y); i is no malized so ha i akes alues be ween 0 (pe ec equali y) and
1 (pe ec inequali y); i has a simple and effec i e in e p e a ion (small alues o his index amoun o ai
income dis ibu ions, whe eas high alues indica e unequal dis ibu ions).
Ano he impo an ins umen o compa e dis ibu ions acco ding o inequali y is he so-called Lo enz
o de ing. Le X1and X2be wo a iables wi h Lo enz cu es 1and 2, espec i ely. I is said ha X1is
less han o equal o X2in he Lo enz o de , w i en X1≤LX2, i 1( ) ≥2( ), o all ∈[0, 1]. In his
case, we ha e ha pe ≥1≥2, whe e pe is he pe ec equali y cu e defined in (8). In o he wo ds,
income is dis ibu ed in a mo e equi able manne in X1 han in X2.
2.2. ROC cu es in machine lea ning
Bina y supe ised classifica ion is one o he main s a is ical echniques in machine lea ning; see Has ie
e al. [17]. In his con ex , we wan o classi y an objec in one o wo g oups labelled 0 and 1. We obse e
a andom ec o (X, Y), whe e Xis he p edic o , usually a mul idimensional (o unc ional) a iable, and
A. Baíllo e al. / J. Ma h. Anal. Appl. 514 (2022) 126335 5
Y∈{0, 1}indica es he g oup membe ship. Usual p ocedu es in machine lea ning combine he in o ma ion
in X o cons uc a sco e o ma ke S o p edic Y. This sco e is ypically an es ima e o he pos e io
p obabili y P(Y=1|X=x)o some inc easing unc ion o his quan i y. We can assume ha membe s o
he class {Y=0}ha e o en smalle alues o he sco e; i no , we can in e change he labels. Then, highe
alues o Sp o ide s onge e idence in a ou o he e en {Y=1}.
We deno e by Fi( ) =P(S≤ |Y=i), i =0, 1and ∈R, he condi ional dis ibu ion unc ions o
he sco e in he g oups. Any easonable x ∈Rcan be used as a cu -off o ob ain a classifica ion ule by
assigning {Y=1}whene e {S>x}, and {Y=0}i {S≤x}. In p ac ice, he selec ed alue o he cu -off
poin x ypically depends on he p io p obabili y o he g oups and misclassifica ion cos s. The p obabili y
o de ec ion o he classifie gene a ed by he h eshold xis
PD(x)=P(S>x|Y=1)=1−F1(x)
and he p obabili y o alse de ec ion is
PFD(x)=P(S>x|Y=0)=1−F0(x).
The unc ions PD(x)and PFD(x)a e also known as hi a e and alse ala m a e, espec i ely.
In his con ex , ROC cu es a e commonly used o e alua e he p edic i e abili y o bina y classifie s.
Fo mally, he ROC cu e is he pa ame ic cu e in [0, 1]2gi en by {(PFD(x), PD(x)) :x ∈R}. I Fi,
i =0, 1a e con inuous and s ic ly inc easing, he ROC cu e is he g aph o he unc ion
( )=1−F1(F−1
0(1 − )), ∈(0,1),
wi h (0) =0and (1) =1.
We obse e ha a good classifie should ha e a high p obabili y o de ec ion and low p obabili y o
alse de ec ion. The e o e, classifie s wi h ROC cu es close o he cons an 1 a e p e e ible. A pe ec
classifie has he ROC cu e (0) =0and (x) =1, x ∈(0, 1], while a andom classifie has a ROC
cu e on he diagonal o [0, 1]2. The e o e, he ROC cu e gi es in o ma ion abou he p ecision o a bina y
classifie . Fu he , ROC cu es a e conca e (see Lloyd [22]) and he a ea unde he ROC cu e (AUC) is
used o e alua e he pe o mance o he classifie . In pa icula , Rain (4)is he se o ROC cu es wi h
AUC sco ing a. A his poin i should be commen ed ha wi hin he machine lea ning communi y ROC
cu es a e conside ed con ex, as hey a e iewed om he line {(x, 1) :x ∈(0, 1]}. We ollow he e he
usual e minology in ma hema ics. Finally, i wo classifie s ha e o de ed ROC cu es, he abo e cu e
co esponds o he classifie ha is uni o mly be e han he o he one; ha is, i gi es be e esul s o
each cu -off poin x ha is selec ed o ca y ou he classifica ion p ocedu e.
3. Ex eme poin s o Laand Ra
The se s Ladefined in (3)(see also (10)) and Rain (4)a e clea ly con ex. The ollowing esul asse s
ha hey a e also compac in L1.
P oposi ion 1. Fo each a ∈[0, 1] and b ∈[1/2, 1], he se s Laand Rba e compac in L1.
As Laand Rba e con ex and compac , hei ex eme poin s acqui e special ele ance. We ecall ha
an ex eme o a con ex se is a poin ha canno be exp essed as a p ope con ex combina ion o o he
poin s wi hin he se . Fo mally, gi en a con ex se C, x ∈Cis an ex eme poin o Ci x = x1+(1 − )x2,
o some ∈(0, 1) and x1, x2∈C, implies ha x1=x2. In he ollowing we deno e by Ex (C) he se o
ex eme poin s o C.
6A. Baíllo e al. / J. Ma h. Anal. Appl. 514 (2022) 126335
Fig. 2. The unc ions a
x1, wi h a =0.5and x1=0.25 (le panel), ma
x2, wi h a =0.5and x2=0.7 (cen al panel) and na
x1,x2, wi h
a =0.5and x1=0.3, x2=0.9 ( igh panel).
The nex heo em de e mines he se o ex eme poin s o La.
Theo em 1. Fo a ∈[0, 1], we ha e ha
Ex (La)=a
x1:x1∈[0,a]∪ma
x2:x2∈(a, 1)∪na
x1,x2:x1∈(0,a),x
2∈(a, 1),
whe e a
x1, ma
x2and na
x1,x2a e he piecewise affine unc ions o Lasuch ha
a
x1:⎧
⎪
⎪
⎨
⎪
⎪
⎩
0→ 0
x1→ 0
1−→ 1−a
1−x1,
ma
x2:⎧
⎪
⎪
⎨
⎪
⎪
⎩
0→ 0
x2→ x2−a
1→ 1,
na
x1,x2:⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
0→ 0
x1→ 0
x2→ x2−a
1−x1
1→ 1
(12)
(wi h he no a ion a
x1(1−) = lim ↑1a
x1( )and he con en ion 1
1=pi in (8)).
Theo em 1summa izes he in o ma ion o La(an infini e-dimensional collec ion) in he se o i s ex eme
poin s, which has only dimension 2. Fu he mo e, as Lais compac in L1, i iden ifies all possible maximize s
o con ex and con inuous unc ionals. To p o e Theo em 1(Sec ion 6) we fi s show ha wice diffe en ia ion
de e mines an affine isomo phism be ween Laand he se o non-nega i e measu es on (0, 1) wi h some
es ic ions. A e wa ds, we iden i y hose combina ions o del a measu es ha a e ex eme poin s.
In Fig. 2we ha e depic ed a ious ex eme poin s o La, wi h a =0.5. The p obabilis ic and economic
meaning o some o hese Lo enz cu es is desc ibed in Sec ion 4.
Rema k 1. Le us conside he map
T(x)=1−(1 −x),∈L,x∈[0,1].(13)
Obse e ha , o a ∈[0, 1], Tdefines a na u al bijec ion be ween Laand R(1+a)/2whose in e se is i sel .
Fu he , o ∈[0, 1] and 1, 2∈L, i holds ha
T( 1+(1− )2)= T(1)+(1− )T(2).
The e o e, Tp ese es ex eme poin s and he se Ex (Ra)is di ec ly ob ained om Theo em 1:
Ex (Ra)={T():∈Ex (L2a−1)}, o a∈[1/2,1].
A. Baíllo e al. / J. Ma h. Anal. Appl. 514 (2022) 126335 7
4. Maximum dis ance be ween Lo enz o ROC cu es
Gi en wo Lo enz (o ROC) cu es wi h fixed Gini indices, in his sec ion we quan i y how “ a ” hey
can be om one ano he . We only conside he p oblem o Lo enz cu es, as he co esponding ques ion
o ROC cu es is analogous (see Rema k 3). We also show ano he p ope y o hose cu es o which he
maximal dis ance is a ained, which is connec ed wi h s ochas ic o de s.
4.1. Compu a ion o he maximum dis ance
We a e in e es ed in compu ing he alue d(La, Lb)(a, b ∈[0, 1]), o a sui able me ic don L ×L, whe e
Lis defined in (1), and Laand Lbas in (3)(see also (10)). Theo em 1is ex emely use ul o his pu pose. I
dis defined h ough a no m, dis a con ex and con inuous unc ional on ( he con ex se ) La×Lb. The e o e,
as long as Laand Lba e compac , by Baue ’s maximum p inciple, he sup emum o don La×Lbis a ained
in Ex (La×Lb) =Ex (La) ×Ex (Lb). Thus, hanks o Theo em 1, we educe he calcula ion o d(La, Lb)
o a fini e-dimensional p oblem.
The exac compu a ion o d(La, Lb) will e en ually depend on he pa icula choice o he me ic d. We
no e ha he Gini coefficien i sel is defined in e ms o a (no malized) L1-dis ance be ween Lo enz cu es;
see o mula (11). This is indeed a sensible and con enien choice o measu e dissimila i ies be ween Lo enz
cu es (and hei associa ed p obabili y dis ibu ions). The L1dis ance be ween Lo enz cu es has also
been used in Zheng [36] ela ed o almos s ochas ic dominance o Leshno and Le y [21]. Explici ly, endow
he se Lin (1)(o , analogously, he se Rin (2)) wi h he Lo enz dis ance
dL(1,
2)= 1−2
pe −pi=21−2=2
1
0|1−2|,
1,
2∈L.(14)
Rema k 2. Le us conside X1, X2 wo posi i e and in eg able andom a iables wi h posi i e expec a ion
and Lo enz cu es 1and 2, espec i ely. We can define d(X1, X2) =dL(1, 2). We ha e ha dis ac ually
a pseudo-me ic (on he space o posi i e and in eg able andom a iables) because d(X1, X2) =0holds i
and only i X1=s cX2, whe e c >0is a cons an and ‘=s ’ s ands o s ochas ic equali y.
By (7), he diame e o Lwi h espec o he me ic dLis
diam(L)=sup{dL(1,
2):1,
2∈L}=dL(pe,
pi)=1.
We u he obse e ha Lain (10)is he se o ∈Lsuch ha dL(, pe) =a. Fo any fixed a, b ∈[0, 1], La
and Lba e compac se s in L1(see P oposi ion 1). The e o e, om Theo em 1, he maximum
M(a, b)=max{dL(1,
2):1∈L
aand 2∈L
b}(15)
is a ained a Ex (La) ×Ex (Lb).
No e ha
M(a, b)= max
1,2∈L 21−2subjec o
1
0
1=1−a
2and
1
0
2=1−b
2.
The e o e, he compu a ion o (15), which in p inciple is an infini e-dimensional con ex maximiza ion p ob-
lem wi h wo linea cons ain s, is educed o a fini e-dimensional op imiza ion p oblem.
8A. Baíllo e al. / J. Ma h. Anal. Appl. 514 (2022) 126335
Fig. 3. The unc ions −
0.4(le panel) and +
0.5( igh panel).
Defini ion 1. We say ha he pai (1, 2) ∈L ×Lis ex emal i
dL(1,
2)=M(G(1),G(2)).
The pai o p obabili y dis ibu ions associa ed o an ex emal pai o Lo enz cu es will also be called
ex emal dis ibu ions.
Fo no a ional con enience, we ename he unc ions a
aand a
0in (12)as −
aand +
a, espec i ely. In o he
wo ds, o 0 ≤a ≤1, −
a, +
a∈L
aa e defined as
−
a( )=max0, −a
1−aand +
a( )=(1 −a) , i 0 ≤ <1,
1,i =1 (16)
(wi h he ag eemen ha −
1≡pi defined in (8)). These wo unc ions will play an essen ial ole in he es
o he sec ion. In Fig. 3we display wo o hese unc ions.
The ollowing heo em, which is he main heo e ical esul o his sec ion, p o ides an explici exp ession
o M(a, b)and shows ha his maximum dis ance is p ecisely a ained a unc ions o he o m (16). The
compu a ion o M(a, b), which begins a Theo em 1and is collec ed in Sec ion 6, e eals ha his issue is
mo e delica e and complex han expec ed.
Theo em 2. Fo 0 ≤a, b ≤1, le M(a, b)be as in (15). We ha e ha
M(a, b)=(1 −a)b2+(1−b)a2
a+b−ab (17)
( he alue M(0, 0) =0is aken by con inui y). Mo eo e , (−
a, +
b)and (+
a, −
b)a e pai s o ex emal Lo enz
cu es wi hin he se La×Lb.
Theo em 2asse s ha he maximum dis ance in (15)is a ained a he pai s (−
a, +
b)and (+
a, −
b).
Hence, he associa ed p obabili y dis ibu ions (unique up o posi i e scale ans o ma ions) a e ex emal.
The unc ion −
ais he Lo enz cu e o a popula ion in which a p opo ion ao he people ha e 0income
and he es , a p opo ion 1 −a, ha e equal and posi i e income. In o he wo ds, (up o posi i e scale
ans o ma ions) −
ais he Lo enz cu e o a a iable X−
awi h Be noulli (1 −a) dis ibu ion, ha is,
P(X−
a=0) =aand P(X−
a=1) =1 −a. On he o he hand, +
bis no a p ope Lo enz cu e, bu i can be
exp essed as he limi (as ngoes o infini y) o Lo enz cu es o popula ions wi h nindi iduals whe e n −1
o hem ai ly sha e a p opo ion (1 −b)o he weal h and he e is only one “lucky pe son” who accumula es
A. Baíllo e al. / J. Ma h. Anal. Appl. 514 (2022) 126335 9
he es o he o al weal h ( he p opo ion b). Hence, +
bcan be ob ained as he limi (as ngoes o infini y)
o he Lo enz cu es o a sequence o andom a iables X+
b(n) aking he alues (1 −b)n/(n −1) (wi h
p obabili y 1 −1/n) and bn (wi h p obabili y 1/n).
The minimum alue o dL(1, 2) o 1∈L
aand 2∈L
bis easily seen o be |b−a|(see Lemma 1in
Sec ion 6). Wi h his and Theo em 2we can see he ange o alues o he dis ance dL(1, 2).
Co olla y 1. Fo 0 ≤a, b ≤1, we ha e ha
{dL(1,
2): 1∈L
aand 2∈L
b}=[|b−a|,M(a, b)] ,
whe e M(a, b)is gi en in (17).
P oo . The minimum and maximum o dL(, m)among ∈L
aand m ∈L
ba e |b−a|and M(a, b), as
calcula ed in Lemma 1and Theo em 2, espec i ely. Now, he se {dL(, m) : ∈L
aand m ∈L
b}is
compac and connec ed as i is he image by he con inuous unc ion dLo he compac and connec ed se
La×Lb(in he space L1). The esul ollows.
Theo em 2also allows us o compu e he maximum dis ance be ween Lo enz cu es wi h a gi en diffe ence
o hei Gini indices.
Co olla y 2. Fo −1 ≤c ≤1, le us conside
M∗(c)=max{M(a, b):a, b ∈[0,1] and b−a=c}.
We ha e ha
M∗(c)=M(ac,a
c+c)wi h ac=(4−c−8+c2)/2 (18)
and
M∗(c)=8−8+(c2+8)
3/2
c2+4 .(19)
Defini ion 2. We say ha he pai (1, 2) ∈L
2is supe -ex emal i
dL(1,
2)=M∗(G(1)−G(2)).
The associa ed pai s o p obabili y dis ibu ions will be also called supe -ex emal dis ibu ions.
Ob iously, each supe -ex emal pai is ex emal because i always holds ha
M(a, b)≤M∗(a−b), o 0 ≤a, b ≤1.
Howe e , om Theo em 2and o any 0 ≤c ≤1, among all he pai s (−
a, +
a+c)and (−
a+c, +
a)(wi h
a ∈[0, 1 −c]) o ex eme Lo enz cu es wi h a alue c o he diffe ence o hei Gini indices he e a e only
wo supe -ex emal cu es. Namely, he pai s co esponding o a =acin (18).
Obse e ha M∗(0) is he maximum possible dis ance be ween Lo enz cu es wi h equal Gini indices.
By Theo em 2and Co olla y 2, we ha e ha he maximum dis ance be ween wo income dis ibu ions bo h
wi h Gini indices equal o ais
16 A. Baíllo e al. / J. Ma h. Anal. Appl. 514 (2022) 126335
Fig. 8. The inequali y index
ˆ
I o Spanish equi alised disposable income co esponding o yea 2008 (X1) and each o he yea s in
he span 2009–2020 (X2).
We finally gi e some insigh on he su p ising alue o
ˆ
I o 2020, ha would indica e a educ ion in
Spanish income inequali y wi h espec o 2008, in spi e o he economic effec s o he Co id-19 pandemic
on he wo ld economy and, pa icula ly, on he Spanish one (see, e.g., IMF [19]). As explained by he INE in
a p ess elease, he LCS collec s in o ma ion om a sample o households ega ding hei li ing condi ions
a he ime o he in e iew ( ou h imes e o he yea ) as well as hei income in he p e ious yea . Thus,
he impac o he pandemic on he LCS-2020 is only pa ially eflec ed.
5.4. A confidence egion o he inequali y index I(1, 2)
In his sec ion, we employ a s anda d boo s ap scheme (see E on [12]), a widely used s a is ical echnique
based on plug-in es ima ion and esampling, combined wi h a non-pa ame ic se es ima ion echnique o
ob ain a confidence egion o he bidimensional ela i e inequali y index Iin (23).
Le xj1, ..., xjnjdeno e he obse a ions om popula ion Xj o j=1, 2. Based on he samples we
cons uc a confidence egion o he popula ion index I(1, 2)a he confidence le el 1 −αin he ollowing
way. Fi s , we ex ac Bboo s ap samples om each o he o iginal samples {xji}nj
i=1, j=1, 2, and we
compu e he co esponding boo s ap e sion o he empi ical inequali y index:
O iginal samples Boo s ap samples Boo s apped indices
x11,...,x
1n1
x21,...,x
2n2
−→
−→
x∗b
11,...,x
∗b
1n1
x∗b
21,...,x
∗b
2n2−→ ˆ
I∗b,b=1,...,B
Boo s ap usually gi es good esul s wi h commonly la ge sample sizes a ailable in inequali y analysis and
machine lea ning se ings. A e ob aining he boo s ap sample
ˆ
I∗1, ...,
ˆ
I∗Bo empi ical inequali y indices,
we use he local con ex hull (LoCoH) (also called k-nea es neighbou con ex hull in he li e a u e o se
es ima ion, see Ge z and Wilme s [16]) o cons uc a confidence egion o I(1, 2). The LoCoH p o ides
esul s ha adap well o he boo s ap sample and o he shape o he egion Δ whe e I akes alues. The
cons uc ion o he LoCoH is as ollows. Fo a fixed in ege k>0, we cons uc he con ex hull o each
ˆ
I∗band i s k−1nea es neighbou s. Then hese hulls a e o de ed acco ding o hei a ea, om smalles
o la ges . The LoCoH is he polygonal egion ha esul s o p og essi ely aking he union o he hulls
A. Baíllo e al. / J. Ma h. Anal. Appl. 514 (2022) 126335 17
Table 1
Values o kand p(k) o Spanish income da a in 2008 and 2019 (in bold he alues o k
and p(k)such ha his p opo ion is closes o 95%).
k100 200 300 400 500 600 700 800 900
p(k0.05) 0.923 0.932 0.935 0.936 0.942 0.933 0.938 0.934 0.938
om he smalles upwa ds, un il a p opo ion 1 −αo
ˆ
I∗b’s is included in he egion. We use he no a ion
ˆ
S(k) = LoCoHk(ˆ
I∗1, ...,
ˆ
I∗B) o deno e he esul ing confidence egion.
Selec ing an adequa e alue o he numbe o neighbou s kis a ele an ma e , as i c i ically affec s he
shape o he LoCoH, bu no au oma ized selec ion p ocedu es ha e ye been conside ed in he li e a u e.
We p opose o use a lea e-one-ou scheme o selec he “op imal” kamong he alues in a g id k1, ..., kM.
The idea is o de e mine, o each b =1, ..., B, he LoCoH
ˆ
S(b)(k) = LoCoHk(ˆ
I∗β, β=1, ..., B, β=b)
based on he sample o boo s apped inequali y indices om which he b- h one has been emo ed. Then,
we compu e he p opo ion o imes ha
ˆ
S(b)(k)con ains he le -ou
ˆ
I∗b
p(k)=
B
b=1
1ˆ
S(b)(k)(ˆ
I∗b)/B.
Ou p oposal o choosing he numbe o neighbou s in he LoCoH is
kα=a gmin
k∈{k1,...,kM}|p(k)−(1 −α)|.(26)
We use he p ocedu e desc ibed abo e o compu e a boo s ap confidence egion o he inequali y index
Ico esponding o wo o he samples conside ed in Sec ion 5.3. We ha e chosen he equi alised disposable
income o Spain in 2008 (n1= 12987) and in 2019 (n2= 15861). The Gini index in his case akes he
alue 32.9% o bo h yea s, and he empi ical bidimensional inequali y index is
ˆ
I=(−10−4, 0.006). As he
componen s o he index a e no bo h close o he o igin (0,0), we can conclude ha he dis ibu ion o
income was no exac ly he same in 2008 and 2019. Indeed, his is confi med by he wo empi ical Lo enz
cu es and hei diffe ence ( e-scaled by he maximum absolu e diffe ence o imp o e he isualiza ion)
plo ed in Fig. 9: in 2019 he poo es hal o he Spanish popula ion had a smalle cumula i e p opo ion o
income han in 2008. Rega ding he p oximi y o he wo Lo enz cu es in Fig. 9(a) and he low alues o
he wo componen s o
ˆ
I, le us no e ha Lo enz cu es o he income o he same coun y in wo diffe en
yea s a e ne e adically diffe en .
We ha e ex ac ed B= 1000 boo s ap samples om each o he da a se s co esponding o 2008 and 2019
and compu ed he esul ing empi ical indices
ˆ
I∗b, b =1, ..., 1000. We ha e de e mined he boo s ap LoCoH
confidence egion
ˆ
S(k0.95), a he confidence le el o 95%, o he ela i e inequali y index Icompa ing 2008
and 2019. The alue o k0.95 was selec ed om he g id (100, 200, ..., 900) ia he lea e-one-ou p ocedu e
desc ibed be o e and acco ding o (26)wi h 1 −α=0.95. Table 1displays he p opo ion p(k)o imes ha
he le -ou index
ˆ
I∗bis con ained in he LoCoH
ˆ
S(b)(k) cons uc ed wi h he emaining 999 indices. The
numbe k0.95 = 500 o nea es neighbou s yielding he p opo ion p(k)nea es o 0.95 esul s in he LoCoH
ˆ
S(500) o Fig. 10. Rema k he con as wi h Fig. 11, whe e we ha e plo ed he boo s ap inequali y indices
co esponding o yea s 2008 and 2016 in Spain and he confidence egion cons uc ed using he LoCoH wi h
k0.95 = 600.
5.5. Hypo hesis es s o he inequali y index I(1, 2)
Once de e mined a confidence egion o I(1, 2)a a le el (1 −α)(as explained in he p e ious sec ion),
we can use i o ca y ou he s a is ical es (wi h significance le el α)
18 A. Baíllo e al. / J. Ma h. Anal. Appl. 514 (2022) 126335
Fig. 9. (a) Empi ical Lo enz cu es,
ˆ
1and
ˆ
2, o he equi alised household income in Spain in 2008 and 2019 espec i ely; (b)
Diffe ence
ˆ
1−ˆ
2 e-scaled by ˆ
1−ˆ
2∞.
Fig. 10. Confidence egion o he bidimensional ela i e inequali y index Io Spain in 2008 and 2019. The c ossed poin in whi e
is he empi ical index
ˆ
I.
H0:I(1,
2)=I0 e sus H1:I(1,
2)=I0,
whe e I0is a known fixed alue in he egion Δgi en in (24). The p ocedu e o es he simple hypo hesis
H0:I=I0 ollows by he duali y be ween confidence egions and hypo hesis es s: we ejec H0a le el α
whene e I0does no belong o he confidence egion o I(1, 2)a a le el (1 −α).
Fo ins ance, he confidence egion a le el 95% in Fig. 10 does no in e sec he le diagonal L2, which
means ha he e is e idence o ejec ha he ela i e inequali y index Icompa ing 2008 and 2019 lies
on any poin o he le diagonal. In o he wo ds, we can affi m (wi h significance le el 5%) ha 2019 did
no dis ibu e income mo e ai ly han 2008. Howe e , o he same yea s, he confidence egion
ˆ
S(500)
in e sec s he igh diagonal L1, so we canno ejec ha he 2019 Lo enz cu e is below ha o 2008. The
si ua ion is e en mo e clea o he yea s 2008 and 2016 (see Fig. 11).
A. Baíllo e al. / J. Ma h. Anal. Appl. 514 (2022) 126335 19
Fig. 11. Confidence egion o he bidimensional ela i e inequali y index Io Spain in 2008 and 2016. The c ossed poin in whi e
is he empi ical index
ˆ
I.
6. P oo s: ex emal poin s and maximum dis ance
He e we collec he p oo s o P oposi ion 1and Theo ems 1and 2. Fi s , we enume a e some egula i y
p ope ies o he unc ions in he se Ldefined in (1) ha will be use ul h oughou his sec ion.
6.1. Regula i y p ope ies and compac ness o La
We s a wi h a sligh change in he defini ion o he unc ions in he se Lo (1). Gi en ∈Lwe
edefine he alue o a 1as (1) =sup
[0,1) . This edefini ion is mo i a ed by he ac ha , as shown in
he ollowing p oposi ion, unc ions in Lbecome con inuous in [0, 1]. In addi ion, he con exi y o Land
La emains ue. Wi h his defini ion, Lbecomes he se o con ex :[0, 1] →[0, 1] such ha (0) =0and
(1) =sup
[0,1) .
In he ollowing p oposi ion we deno e by W1,1(0, 1) he Sobole space W1,1in he in e al (0, 1), which
is equi alen o he se o absolu ely con inuous unc ions in [0, 1]; i is endowed wi h he no m
W1,1(0,1) =+,
whe e is he dis ibu ional de i a i e o , which coincides a.e. wi h he de i a i e o . Fo α∈(0, 1)
we deno e by W1,∞(0, α) he Sobole space W1,∞in he in e al (0, α), which is equi alen o he se o
Lipschi z con inuous unc ions in [0, α]; i is endowed wi h he no m
W1,∞(0,α)=sup
(0,α)||+ ess sup
(0,α)||.
See, e.g., B ezis [6, Chap e 8] o E ans and Ga iepy [14, Chap e 4] o he defini ion and p ope ies o
hese spaces.
P oposi ion 5. Le a ∈[0, 1] and ∈L
a. Then
(a) The unc ion is non-dec easing, absolu ely con inuous in [0, 1] and Lipschi z in [0, α] o each α∈(0, 1).
Mo eo e ,
W1,1(0,1) ≤1+1−a
2,
20 A. Baíllo e al. / J. Ma h. Anal. Appl. 514 (2022) 126335
and o each α∈(0, 1),
W1,∞(0,α)≤1+ 1−a
(1 −α)2.
(b) The unc ion is locally o bounded a ia ion, he igh de i a i e (x+)exis s o all x ∈[0, 1) and is
non-dec easing. Mo eo e , (0+) ≥0.
(c) is a non-nega i e Radon measu e.
P oo . Con ex unc ions a e locally Lipschi z (see E ans and Ga iepy [14, Theo em 6.3.1] o Simon [30,
Theo em 1.19]), ha e a fi s de i a i e locally o bounded a ia ion (see E ans and Ga iepy [14, Theo em
6.3.3]) and ha e a second de i a i e in he sense o dis ibu ions (see E ans and Ga iepy [14, Theo em 6.3.2]
o Simon [30, Theo em 1.29]), which in ac is a non-nega i e Radon measu e. Fu he , he igh de i a i e
(x+), exis s o all x ∈[0, 1) and is non-dec easing (see Simon [30, Theo em 1.26]). As (0) =0and ≥0,
we necessa ily ha e ha (0+) ≥0, so (x+) ≥0 o all x ∈[0, 1). By he e sion o he undamen al
heo em o calculus o con ex unc ions (see Simon [30, Theo em 1.28]), is non-dec easing. In addi ion,
he de i a i e o exis s a.e. and coincides a.e. wi h he igh de i a i e, so ≥0a.e. In pa icula ,
=
1
0
( )d =(1) −(0) ≤1.
We conclude ha W1,1(0,1) ≤1 +1−a
2.
On he o he hand, we obse e ha he affine unc ion s :[0, 1] →Rgi en by s(x) =(α+)(x −α) +(α)
is a suppo ing line o a he poin (α, (α)). By con exi y, we hence ha e ha s ≤and hen,
1−a
2=≥
1
α
( )d ≥
1
α
s( )d =(α+)(1 −α)2
2+(α)(1 −α)≥(α+)(1 −α)2
2.
We conclude ha
ess sup
(0,α)
≤(α+)≤1−a
(1 −α)2.
Consequen ly, W1,∞(0,α)≤1 +1−a
(1−α)2.
The se Lais clea ly con ex. We a e now eady o p o e P oposi ion 1.
P oo o P oposi ion1in Sec ion 3(Compac ness o Laand Rb). Le {n}n∈Nbe a sequence in La. By
P oposi ion 5, {n}n∈Nis bounded in W1,1(0, 1), so by he Rellich–Kond acho heo em (see B ezis [6,
Theo em 8.8]), he e exis s a subsequence (no elabelled) and an ∈L1such ha n→in L1as n →∞.
This also implies ha G() =a.
On he o he hand, o each α∈(0, 1) we ha e by P oposi ion 5(a) ha {n}n∈Nis bounded in
W1,∞(0, α), so by he Ascoli–A zelà heo em (see B ezis [6, Theo ems 4.25 and 8.8]), o a u he subse-
quence, n→uni o mly in [0, α]as n →∞. In pa icula , (0) =0and 0 ≤ ≤1in [0, α]. As he poin wise
limi o con ex unc ion is a con ex unc ion, we ob ain ha is con ex in [0, α]. The e o e, 0 ≤ ≤1in
[0, 1) and is con ex in [0, 1). We edefine (1) as (1) =sup
[0,1), so ha becomes con inuous in [0, 1]. We
also ob ain ha 0 ≤ ≤1in [0, 1] and is con ex in [0, 1]. The e o e, ∈L
aand he p oo is finished.
A. Baíllo e al. / J. Ma h. Anal. Appl. 514 (2022) 126335 21
6.2. P oo o Theo em 1in Sec ion 3(ex eme poin s o La)
We will use an al e na i e desc ip ion o he elemen s in Lain e ms o posi i e measu es concen a ed on
he in e al (0, 1). The main idea is based on he ollowing ac : any cu e ∈L
ais uni ocally de e mined by
i s second de i a i e, , oge he wi h he condi ions (0) =0and G() =a(o , equi alen ly, =1−a
2).
Gi en a ∈[0, 1], we deno e by Ma he se o non-nega i e Radon measu es μconcen a ed on he in e al
(0, 1) and such ha
1
0
(1 −s)2dμ(s)≤1−aand
1
0
s(1 −s)dμ(s)≤a. (27)
P oposi ion 6. Fo a ∈[0, 1], he map Ta:La→M
adefined by Ta() = is an affine isomo phism wi h
in e se T−1
a:Ma→L
agi en by
T−1
aμ(x)=⎡
⎣1−a−
1
0
(1 −s)2dμ(s)⎤
⎦x+
x
0
(x−s)dμ(s),x∈[0,1].(28)
P oo . Fi s we see ha he map Tais well defined. Gi en ∈L
a, we ha e om P oposi ion 5 ha is a
non-nega i e Radon measu e. As is locally o bounded a ia ion,
( )=(0+)+
0
d(s),a.e. ∈(0,1).
As is locally Lipschi z and (0) =0, o x ∈[0, 1], we ha e ha
(x)=
x
0
( )d =
x
0
⎡
⎣(0+)+
0
d(s)⎤
⎦d
=(0+)x+
x
0
(x−s)d(s),
(29)
whe e o he las equali y we ha e used Fubini’s heo em. In eg a ing in x ∈(0, 1) equali y (29)(and by
Fubini’s heo em again) we ob ain he es ic ion
1−a
2==(0+)
2+1
2
1
0
(1 −s)2d(s).(30)
As (0+) ≥0, om (30)we di ec ly ob ain he fi s inequali y o (27). On he o he hand, (29)and (30)
show ha
(x)=⎡
⎣1−a−
1
0
(1 −s)2d(s)⎤
⎦x+
x
0
(x−s)d(s),x∈[0,1].(31)
Hence, imposing (1) ≤1we ha e he second inequali y o (27).
Now, o μ ∈M
a, we define as in he igh -hand side o (28)and we will check ha ∈L
a. Fi s ,
(0) =0and
22 A. Baíllo e al. / J. Ma h. Anal. Appl. 514 (2022) 126335
1
0
(x)dx=1
2⎡
⎣1−a−
1
0
(1 −s)2dμ(s)⎤
⎦+
1
0
1
s
(x−s)dxdμ(s)=1−a
2.
Fu he , hanks o he second inequali y o (27), we ob ain ha
(1) = 1 −a+
1
0
s(1 −s)dμ(s)≤1.
By Leibniz in eg al ule (diffe en ia ion unde he in eg al sign), i can also be checked ha
(x)=1−a−
1
0
(1 −s)2dμ(s)+
x
0
dμ(s),a.e. x∈(0,1) (32)
and hence
=μas measu es. (33)
As μis posi i e, om (32)we ha e ha is essen ially non-dec easing, is con ex and
(x)≥1−a−
1
0
(1 −s)2dμ(s)≥0,a.e. x∈(0,1),
by he fi s inequali y o (27), so is non-dec easing. In pa icula , ≥0. This shows ha ∈L
a.
Finally, we p o e ha he maps Taand (28)a e mu ually in e se. Gi en ∈L
a, i we apply fi s Taand
hen (28)we ge back hanks o (31). Con e sely, gi en μ ∈M
a, i we apply fi s (28)and hen Tawe
eco e μby (33). Since Tais affine, he p oo is concluded.
Nex we calcula e Ex (Ma). We deno e by δx he Di ac measu e a x ∈[0, 1].
P oposi ion 7. Fo a ∈[0, 1], we ha e ha
Ex (Ma)={0}∪1−a
(1 −x1)2δx1:x1∈(0,a]∪a
x2(1 −x2)δx2:x2∈(a, 1)
∪x2−a
(1 −x1)(x2−x1)δx1+a−x1
(x2−x1)(1 −x2)δx2:x1∈(0,a),x
2∈(a, 1).
P oo . The p oo is di ided in o se e al smalle esul s.
S ep 1: The null measu e μ ≡0 ∈Ex (Ma).
This is di ec as all he measu es in Maa e non-nega i e.
S ep 2: Fo all x1∈(0, a], he measu e μ =1−a
(1−x1)2δx1∈Ex (Ma).
Clea ly, μ ∈M
a. Assume ha μ = 1μ1+ 2μ2, o some 1, 2>0wi h 1+ 2=1and μ1, μ2∈M
a.
Then μi=βiδx1and, due o (27),
βi≥0,β
i(1 −x1)2≤1−a, βix1(1 −x1)≤a, i =1,2.
A. Baíllo e al. / J. Ma h. Anal. Appl. 514 (2022) 126335 23
Thus,
1−a
(1 −x1)2= 1β1+ 2β2≤ 1
1−a
(1 −x1)2+ 2
1−a
(1 −x1)2=1−a
(1 −x1)2,
so, necessa ily, β1=β2=1−a
(1−x1)2, and, hence, μ1=μ2. The e o e, μ ∈Ex (Ma).
S ep 3: Fo all x2∈(a, 1), he measu e
a
x2(1−x2)δx2∈Ex (Ma).
The p oo is simila o he one o he p e ious s ep and i is he e o e omi ed.
S ep 4: I o some x1∈(0, a]and α∈R {0,
1−a
(1−x1)2}, μ =αδx1∈M
a, hen μ /∈Ex (Ma).
The ac μ ∈M
aimplies ha 0 <α< 1−a
(1−x1)2. The e o e, o ε >0small enough, we ha e ha
μ ±εδx1∈M
asince bo h a e posi i e measu es and he es ic ions (27)a e sa isfied; indeed,
1
0
(1 −s)2d(μ±εδx1)(s)=(α±ε)(1 −x1)2<1−a
and
1
0
s(1 −s)d(μ±εδx1)=(α±ε)x1(1 −x1)<1−a
1−x1
x1≤a.
Finally, we can w i e μ =1
2(μ +εδx1) +1
2(μ −εδx1), and, hence, μ /∈Ex (Ma).
S ep 5: I o some x2∈(a, 1) and α∈R {0,
a
x2(1−x2)}, μ =αδx2∈M
a, hen μ /∈Ex (Ma).
The p oo is simila o ha o S ep 4 and i is le o he eade .
S ep 6: Fo all x1∈(0, a)and x2∈(a, 1), μ =x2−a
(1−x1)(x2−x1)δx1+a−x1
(x2−x1)(1−x2)δx2∈Ex (Ma).
I is immedia e o check ha
1
0
(1 −s)2dμ(s)=1−aand
1
0
s(1 −s)dμ(s)=a.
The e o e, μ ∈M
a. Mo eo e , i μ = 1μ1+ 2μ2 o some 1, 2>0wi h 1+ 2=1and μ1, μ2∈M
a, hen
1
0
(1 −s)2dμi(s)=1−aand
1
0
s(1 −s)dμi(s)=a, i =1,2.
Fu he mo e, as μi=2
j=1 βijδxj o some βij ≥0( o i, j=1, 2), we ha e ha
2
j=1
βij(1 −xj)2=1−aand
2
j=1
βijxj(1 −xj)=a, i =1,2,
and, hence,
24 A. Baíllo e al. / J. Ma h. Anal. Appl. 514 (2022) 126335
βi1=x2−a
(1 −x1)(x2−x1),β
i2=a−x1
(x2−x1)(1 −x2),i=1,2.
The e o e, μ1=μ2. Consequen ly, μ ∈Ex (Ma).
S ep 7: I o some α1, α2>0and 0 <x
1<x
2<1wi h
α1=x2−a
(1 −x1)(x2−x1)o α2=a−x1
(x2−x1)(1 −x2)
he measu e μ =2
i=1 αiδxi∈M
a, hen μ /∈Ex (Ma).
By (27), we ha e ha
2
i=1
αi(1 −xi)2≤1−aand
2
i=1
αixi(1 −xi)≤a. (34)
I bo h inequali ies in (34)we e equali ies, we necessa ily ha e ha
α1=x2−a
(1 −x1)(x2−x1)and α2=a−x1
(x2−x1)(1 −x2),
agains ou assump ion. The e o e, a leas one o he wo inequali ies o (34)is s ic . I 2
i=1 αi(1 −xi)2<
1 −a, hen we conside he signed measu e defined by
μ0=x2(1 −x2)δx1−x1(1 −x1)δx2.
Then, i is s aigh o wa d o check ha , o small enough ε >0, μ ±εμ0∈M
aand
μ=1
2(μ+εμ0)+1
2(μ−εμ0).(35)
The e o e, μ /∈Ex (Ma).
I , ins ead, 2
i=1 αixi(1 −xi)2<a, we hen conside he signed measu e
μ0=(1−x2)2δx1−(1 −x1)2δx2.
Again, we ha e ha , o small enough ε >0, μ ±εμ0∈M
aand equali y (35)holds. We conclude ha
μ /∈Ex (Ma).
S ep 8: I μ ∈M
ais suppo ed in mo e han wo poin s, hen μ /∈Ex (Ma).
In his case, he e exis Bo el disjoin se s Ai⊂(0, 1) such ha μ|Ai=0, o i =1, 2, 3. Le (α1, α2, α3) ∈
R3 {(0, 0, 0)}be such ha
3
i=1
αi
Ai
(1 −s)2dμ(s)=0 and
3
i=1
αi
Ai
s(1 −s)2dμ(s)=0.
We conside he signed measu e μ0=3
i=1 αiμ|Ai. Fo ε >0small enough, define μ+and μ−as μ±=
μ ±εμ0. Then μ =1
2μ++1
2μ−wi h μ±=μ. Mo eo e , μ±a e posi i e measu es since
A. Baíllo e al. / J. Ma h. Anal. Appl. 514 (2022) 126335 25
μ±=μ|Ac
1∩Ac
2∩Ac
3+
3
i=1
(1 ±εαi)μ|Ai,
whe e Acs ands o he complemen o he se Ain (0, 1). In ac , μ±∈M
asince
1
0
(1 −s)2dμ±(s)=
1
0
(1 −s)2dμ(s)≤1−aand
1
0
s(1 −s)dμ±(s)=
1
0
s(1 −s)dμ(s)≤a.
The e o e, we conclude ha μ /∈Ex (Ma).
The eigh s eps abo e comple e he p oo .
S ep 8 o he p e ious p oo is ela ed o he wo ks by Winkle [33]and Pinelis [26], whe e hey analyze
he se o ex eme poin s o subse o measu es defined h ough some inequali ies.
P oo o Theo em 1in Sec ion 3(Ex eme poin s o La). By P oposi ion 6, we ha e he equali y
Ex (La)=T−1
a(Ex (Ma)).
Now, we can use P oposi ion 7 o de e mine he se Ex (La). By (28)and P oposi ion 7, we ob ain h ee
amilies o ex eme cu es in La. Fi s , o x1∈(0, a], le a
x1=T−1
a1−a
(1−x1)2δx1and a
0=T−1
a(0). Mo e
explici ly, we ha e ha , o x1∈[0, a]
a
x1(x)= 1−a
(1 −x1)2max{0,x−x1},x∈[0,1].
Second, o x2∈(a, 1), we se ma
x2=T−1
aa
x2(1−x2)δx2. We ob ain ha
ma
x2(x)= 1
x2(x2−a)x+a
1−x2
max{0,x−x2},x∈[0,1].
Finally, o x1∈(0, a)and x2∈(a, 1), le na
x1,x2=T−1
ax2−a
(1−x1)(x2−x1)δx1+a−x1
(x2−x1)(1−x2)δx2. In his case
we ha e ha
na
x1,x2(x)= 1
x2−x1x2−a
1−x1
max{0,x−x1}+a−x1
1−x2
max{0,x−x2},x∈[0,1].
These cu es admi he cha ac e iza ion as piecewise affine unc ions gi en in (12).
The e o e, he p oo o Theo em 1is comple e.
6.3. P oo o Theo em 2in Sec ion 4(maximal dis ance)
He e we compu e he exac alue o M(a, b)in (15). The p oo o his heo em is long and we ha e di ided
i in o se e al esul s. I is based on ollowing p oposi ion.
P oposi ion 8. Fo a, b ∈[0, 1], le M(a, b)be defined in (15). We ha e ha
M(a, b)=max{dL(1,
2):1∈Ex (La)and 2∈Ex (Lb)}.(36)
32 A. Baíllo e al. / J. Ma h. Anal. Appl. 514 (2022) 126335
A 1−√1−a, 1+b
2!=4√1−a−1b+a(b−2) −4√1−a+b2+4
b,
which is seen o be less han he alue o dL(−
a, +
b) (compu ed in Lemma 3), hanks o he es ic ion
a +b2<2b.
A e ha ing checked he alue o Aa he c i ical poin s in he in e io o egion (44), we analyze he
alue o Aon he bounda y. Desc ibing he bounda y o egion (44)is cumbe some since i in ol es se e al
cases, acco ding o he alues o a, b. In any case, he bounda y is clea ly con ained in he se
(x1,x
2)∈[0,a]×[b, 1] : x1=0o x1=ao x2=bo x2=1
o (1 −a)(x2−x1)−(x2−b)(1 −x1)+x1(1 −x1)(x2−b)=0
.
When (1 −a)(x2−x1) −(x2−b)(1 −x1) +x1(1 −x1)(x2−b) =0 he cu es a
x1and mb
x2do no ha e a
p ope c ossing, so, by Lemma 1, A =|b −a|, which does no elease a maximum. The e o e, we a e led o
he maximiza ion o A(x1, x2)in he se
(x1,x
2)∈[0,a]×[b, 1] : x1=0o x1=ao x2=bo x2=1
The alue o Awhen x1=0is
A(0,x
2)=a+b−2ab
a+b−ax2
,
which is dec easing in x2, so he maximum is a ained a x2=band equals
A(0,b)=a+b−2ab
a+b−ab =(1 −a)b2+(1−b)a2
a+b−ab =dL(−
a,
+
b).
The alue o Awhen x1=ais
A(a, x2)=a+b−2ab
ax2+b−ab
which is inc easing in x2, so he maximum is a ained a x2=1and equals
A(a, 1) = (1 −a)b2+(1−b)a2
a+b−ab =dL(−
a,
+
b).
The alue o Awhen x2=bis
A(x1,b)=a2b−a2+ab2−b2+(−4ab +2a+2b)x1+(a+b−2) x2
1
ab −a−b+2x1−x2
1
,
which is dec easing in x1, so he maximum is a ained a x1=0and equals
A(0,b)=(1 −a)b2+(1−b)a2
a+b−ab =dL(−
a,
+
b).
The alue o Awhen x2=1is
A(x1,1) = a2b−a2+ab −b2+a+2b2−3abx1+a−b2x2
1+(b−1) x3
1
ab −a−b+2x1−x2
1
,
A. Baíllo e al. / J. Ma h. Anal. Appl. 514 (2022) 126335 33
which is inc easing in x1. The e o e, he maximum is a ained a x1=aand equals
A(a, 1) = (1 −a)b2+(1−b)a2
a+b−ab .
This concludes he p oo .
We finally obse e ha he p oo o Theo em 2di ec ly ollows om Lemmas 3, 5, 6, and 7.
Acknowledgmen s
A. Baíllo and J. Cá camo a e suppo ed by he Spanish MCyT g an PID2019-109387GB-I00. C. Mo a-
Co al is suppo ed by he Spanish MCyT g an MTM2017-85934-C3-2-P.
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