a Xi :1410.2556 2 [ma h.FA] 13 Sep 2016
ROGERS-SHEPHARD INEQUALITY FOR LOG-CONCAVE
FUNCTIONS
DAVID ALONSO-GUTI´
ERREZ, BERNARDO GONZ´
ALEZ MERINO, C. HUGO JIM´
ENEZ,
RAFAEL VILLA
Abs ac . In his pape we p o e di e en unc ional inequali ies ex ending
he classical Roge s-Shepha d inequali ies o con ex bodies. The o iginal
inequali ies p o ide an op imal ela ion be ween he olume o a con ex body
and he olume o se e al symme iza ions o he body, such as, i s di e ence
body. We cha ac e ize he equali y cases in all hese inequali ies. Ou me hod
is based on he ex ension o he no ion o a con olu ion body o wo con ex
se s o any pai o log-conca e unc ions and he s udy o some geome ical
p ope ies o hese new se s.
1. In oduc ion
A measu e µon Rnis log-conca e i o any measu able se s A, B ⊂Rnand
0< λ < 1,
µ(λA + (1 −λ)B)≥µ(A)λµ(B)1−λ
whene e A, B ⊂Rnand λA + (1 −λ)Ba e measu able, whe e A+B={a+b:
a∈A, b ∈B}is he Minkowski sum.
Log-conca e measu es na u ally appea in Con ex Geome y, since he B unn
Minkowski inequali y es ablishes he log-conca i y o he Lebesgue measu e e-
s ic ed o con ex se s, and o he ma ginal sec ions o con ex se s.
A unc ion :Rn→[0,+∞) is log-conca e i (x) = e−u(x) o some con ex
unc ion u:Rn→(−∞,∞]. As was shown in [16], a measu e µon Rnwi h ull-
dimensional suppo is log-conca e i and only i i has a log-conca e densi y wi h
espec o he Lebesgue measu e.
The class o log-conca e unc ions has p o en o be o g ea impo ance in se e al
a eas o ma hema ics. F om a unc ional poin o iew i has been shown hey
esemble Gaussian unc ions in many di e en ways. Many unc ional inequali ies
sa is ied by Gaussian unc ions, like Poinca e and Log-Sobole inequali ies, also
hold in a mo e gene al subclass o log-conca e unc ions [8, 9, 12]. They also appea
in a eas as In o ma ion Theo y, in he s udy o some impo an pa ame e s, such as
he classical en opy [14]. The e a e many examples in he li e a u e o unc ional
inequali ies wi h a geome ic coun e pa ; P ekopa-Leindle /B unn-Minkowski [27]
and Sobole /Pe y p ojec ion [34] inequali ies a e wo o he main examples. This
has gene a ed an inc easing in e es o ex end se e al impo an pa ame e s o
con ex bodies o unc ional pa ame e s [5, 6, 12, 19, 22, 24] in he class o log-
conca e unc ions.
Da e: Sep embe 14, 2016.
2010 Ma hema ics Subjec Classi ica ion. P ima y 52A20, Seconda y 39B62,46N10.
Key wo ds and ph ases. Roge s-Shepha d inequali y, log-conca e measu es, log-conca e unc-
ions, con olu ion body, geome ic inequali ies.
1
2 DAVID ALONSO, BERNARDO GONZ´
ALEZ, C. HUGO JIM´
ENEZ, RAFAEL VILLA
The class o log-conca e unc ions is o en ega ded as he na u al ex ension
o con ex bodies, aking in o accoun ha he cha ac e is ic unc ion o a con ex
body is a log-conca e unc ion and ha his is he smalles closed unde limi s
class o unc ions ha con ains he densi ies o n-dimensional ma ginals o uni o m
p obabili ies on con ex bodies o highe dimension (we e e o he nex sec ion o
p ecise de ini ions).
In his wo k we ex end Roge s-Shepha d inequali y [28, 29] o he class o log-
conca e unc ions (Theo ems 2.1 and 2.3), cha ac e izing he equali y cases as well.
We p o ide o he unc ional e sions o inequali ies a ound Roge s-Shepha d’s wi h
hei espec i e cha ac e iza ion o equali y cases. While he B unn-Minkowski in-
equali y is commonly seen as he backbone o mode n Con ex Geome y, Roge s-
Shepha d inequali y can be conside ed as a e e se o m o B unn-Minkowski in-
equali y ha no only desc ibes a ela ion be ween Minkowski addi ion and olume,
bu also deals wi h ye ano he undamen al p ope y in con exi y: symme y. The
a eaching in luence o his inequali y becomes e iden as i can be ound as an
ing edien no only in many impo an wo ks in classical and asymp o ic con ex
geome y [11, 23, 21, 26] bu also in many o he s wi h a mo e analy ical la o
[31, 7, 33, 25] and i s ex ension o a unc ional se ing as well as o he en opy o
con ex measu es has al eady been conside ed o ins ance in [17] o [15].
The pape is o ganised as ollows: In Sec ion 2 we p o ide he no a ion used
in he es o he pape and some p e ious esul s and s a e he p ecise esul s we
a e going o p o e. In Sec ion 3 we in oduce he (θ, )-con olu ion bodies o log-
conca e unc ions and p o e he ex ension o Roge s-Shepha d inequali ies (8) and
(9). In Sec ion 4 we in oduce he mo e gene al concep o k- h (θ, )-con olu ion
bodies and p o e a Roge s-Shepha d ype inequali y (10) o su ace a ea. When
pa icula izing o k=nwe ob ain he p e iously in oduced (θ, )-con olu ion
bodies. In Sec ion 5 we cha ac e ize he equali y cases in hese inequali ies. Finally,
in Sec ion 6, we e isi ano he esul a ound Roge s-Shepha d inequali y [17] by
gi ing an ex ended e sion o wo di e en unc ions. Thus, we ex end inequali y
(7) o any wo log-conca e unc ions and cha ac e ize he equali y cases. Since his
inequali y will s eng hen ano he well known Roge s-Shepha d inequali y (4) we
will make use o hese new ools o cha ac e ize he equali y cases in inequali y (4).
2. No a ion and p e ious esul s
The no a ion used in his pape is qui e s anda d in mode n con ex geome y
and consis en wi h o example [32] and [20]. A con ex body is a subse o Rn ha
is con ex, compac and has non-emp y in e io . I is said o be cen ally symme ic
i o any x∈Kwe ha e ha also −x∈K. When s udying geome ic p ope ies
o a con ex body Ki is usually e y con enien o cons uc ano he con ex body
om Kwhich is cen ally symme ic. The e a e many ways o cons uc such a
symme iza ion. One o hem is he so called di e ence body o K, which is he
Minkowski sum o Kand −K. Le us ecall ha he Minkowski sum o wo con ex
bodies Kand Lis de ined as
K+L={x+y∈Rn:x∈K, y ∈L}
={x∈Rn:K∩(x−L)6=∅}.
B unn-Minkowski inequali y (see, o ins ance, [4] o se e al p oo s and cha ac-
e iza ion o he equali y cases) gi es he ollowing lowe bound o he olume o
ROGERS-SHEPHARD INEQUALITY FOR LOG-CONCAVE FUNCTIONS 3
he Minkowski sum o any wo con ex bodies K, L ⊆Rn:
|K+L|1
n≥ |K|1
n+|L|1
n.
As a consequence o B unn-Minkowski inequali y he ollowing ela ion be ween
he olume o he di e ence body K−Kand he olume o Kis always ue
|K−K| ≥ 2n|K|,
wi h equali y i and only i Kis symme ic. In [28] Roge s and Shepha d p o ed a
e e se inequali y. Namely, Roge s-Shepha d inequali y s a es ha o any con ex
body K⊆Rn, we ha e
(1) |K−K| ≤ 2n
n|K|,
wi h equali y i and only i Kis a simplex. This inequali y was ex ended o any
pai o con ex bodies K, L ⊆Rnin [29], showing ha
(2) max
x0∈Rn|K∩(x0−L)||K+L| ≤ 2n
n|K||L|,
wi h equali y i and only i K=−Lis a simplex (see [2] o he cha ac e iza ion o
equali y).
In he same pape [29] he au ho s also conside ed di e en ypes o symme iza-
ion o a con ex body Kand p o ed olume inequali ies o hem. In pa icula
i was shown ha o any con ex body K⊆Rncon aining 0, he olume o he
con ex hull o Kand −K e i ies
(3) |con {K, −K}| ≤ 2n|K|
wi h equali y i and only i Kis a simplex and 0 is one o i s e ices. In he same
pape he au ho s ema ked ha , wi h a simila p oo , he la e inequali y can be
ex ended o he ollowing inequali y o any wo bodies Kand Lcon aining he
o igin
(4) |K∩L||con {K, −L}|≤ 2n|K||L|
and hey sugges ed ha i is likely ha equali y is a ained i and only i K=Lis
a simplex and 0 is one o i s e ices.
Ve y ecen ly, in [2], he olume o he θ-con olu ion bodies K+θLwas s udied,
whe e
(5) K+θL={x∈K+L:|K∩(x−L)| ≥ θmax
z∈Rn|K∩(z−L)|}.
As a consequence o he olume inequali ies ob ained o con olu ion bodies, in-
equali y (2) was eco e ed and he equali y cases we e cha ac e ized.
In [1], simila inclusion ela ions and olume inequali ies we e ob ained o he
h, θ-con olu ion bodies o Kand L, de ined as
K+h,θ L={x∈K+L:h(K∩(x−L)) ≥θmax
z∈Rnh(K∩(z−L))},
whe e his a unc ion sa is ying some p ope ies. As a pa icula case we ha e he
k- h θ-con olu ion bodies o wo con ex bodies, de ined as
K+k,θ L={x∈K+L:Wn−k(K∩(x−L)) ≥θmax
z∈RnWn−k(K∩(z−L))},
4 DAVID ALONSO, BERNARDO GONZ´
ALEZ, C. HUGO JIM´
ENEZ, RAFAEL VILLA
whe e Wn−kdeno es he (n−k)- h que maßin eg al o a con ex body which, by
Kubo a’s o mula (c . [32, p. 295]), can be exp essed as an a e age o he olumes
o he k-dimensional p ojec ions o K
Wn−k(K) = |Bn
2|
|Bk
2|ZGn,k |PE(K)|dµ(E).
(Gn,k deno es he se o k-dimensional linea subspaces, dµ is he Haa p obabili y
measu e on Gn,k and PE(K) is he p ojec ion o Kon a subspace E). As a conse-
quence o hese olume inequali ies he ollowing Roge s-Shepha d ype inequali y
o any wo con ex bodies K, L ⊆Rn, which in ol es he su ace a ea o Kand L,
is ob ained
(6) |K+L| ≤ 2n
n|K||∂L|+|L||∂K|
2 maxx0∈Rn|∂(K∩(x0−L))|,
whe e |∂K|is he su ace a ea o K. No ice ha when L=−Kwe eco e inequal-
i y (1). Le us ecall ha , up o a cons an which depends only on he dimension
n, he su ace a ea o a con ex body Kequals he que maßin eg al W1(K).
Inequali y (3) was ex ended o he con ex o log-conca e unc ions in [17], whe e
he au ho p o ed ha o any log-conca e unc ion , i i s di e ence unc ion is
de ined by
∆ (z) = sup np (x) (−y) : x, y ∈Rn: 2z=x+yo,
hen
(7) ZRn
∆ (x)dx ≤2nZRn
(x)dx.
Taking (x) = e−hK◦(x), wi h hK◦(x) = maxy∈K◦hx, yi he suppo unc ion o he
pola se o a con ex body Kcon aining he o igin, inequali y (3) is eco e ed. This
inequali y was also ex ended in [3].
In his pape we ex end inequali ies (1), (2) and (6) o log-conca e unc ions.
Be o e we s a e ou esul s we need o in oduce some mo e no a ion.
Gi en , g wo log-conca e unc ions, hei con olu ion de ined by
∗g(x) = ZRn
(z)g(x−z)dz
is also a log-conca e unc ion in Rn. I (x) = χK(x) and g(x) = χL(x) a e he
cha ac e is ic unc ions o wo con ex bodies, hen ∗g(x) = |K∩(x−L)|.
The Asplund p oduc o wo log-conca e unc ions is de ined by
⋆ g(x) = sup
z∈Rn
(z)g(x−z).
I (x) = χK(x) and g(x) = χL(x), hen ⋆ g(x) = χK+L(x). This ope a ion
is he na u al ex ension o he Minkowski sum o con ex bodies, as i has been
shown when ex ending geome ic inequali ies o he con ex o gene al log-conca e
unc ions (see o ins ance [5]).
Rema k. No ice ha i bo h and ga e in eg able and con inuous when es ic ed
o hei suppo s hen his sup emum is a maximum since, in such case, i ⋆g(x) =
0, hen o any z∈Rnwe ha e ha (z)g(x−z) = 0 and i ⋆ g(x)>0, hen
he e exis s a > 0 such ha he se
A (x) := {z∈supp ∩(x−supp g) : (z)g(x−z)≥ k k∞kgk∞}
ROGERS-SHEPHARD INEQUALITY FOR LOG-CONCAVE FUNCTIONS 5
is no emp y. Since and ga e in eg able log-conca e unc ions his se is con ex
and bounded. Thus, i s closu e is a compac con ex se . Since bo h and ga e
con inuous when es ic ed o hei suppo s and
⋆ g(x) = sup
z∈Rn
(z)g(x−z) = sup
z∈cl(A (x))
(z)g(x−z)
he unc ion (z)g(x−z) is con inuous on he compac se A (x) and he maximum
is a ained.
Wi h his no a ion, we p o e he ollowing ex ension o inequali y (2).
Theo em 2.1. Le , g :Rn→Rbe wo in eg able log-conca e unc ions wi h
ull-dimensional suppo such ha and ga e con inuous when es ic ed o hei
suppo s. Then
(8) k ∗gk∞ZRn
⋆ g(x)dx ≤2n
nk k∞kgk∞ZRn
(x)dx ZRn
g(x)dx.
Fu he mo e, his inequali y becomes an equali y i and only i (x)
k k∞=g(−x)
kgk∞is he
cha ac e is ic unc ion o an n-dimensional simplex.
In case we conside g(x) = (−x) he la e inequali y can be imp o ed o he
ollowing ex ension o inequali y (1):
Theo em 2.2. Le be a log-conca e unc ion wi h ull-dimensional suppo and
con inuous when es ic ed o i and le ¯
(x) = (−x). Then
(9) ZRn
⋆ ¯
(x)dx ≤2n
nk k∞ZRn
(x)dx.
Fu he mo e, his inequali y becomes an equali y i and only i (x)
k k∞is he cha ac-
e is ic unc ion o an n-dimensional simplex.
Le us men ion ha he p e ious inequali y was i s ob ained by Colesan i
[17, Theo em 4.3] whe e he au ho p o es i in he quasi-conca e case wi hou
cha ac e izing he equali y case.
The ac ha inequali y (9) is an imp o emen o inequali y (8) ollows om
Young’s inequali y k ∗¯
k∞≤ k k1k k∞.
The no ion o que maßin eg als has also been ex ended om con ex bodies o
he se ing o log-conca e unc ions. In [18], [22] and [30], he case o he pe ime e
and he mean wid h is conside ed while, in [13], a di e en de ini ion is gi en o all
he que maßin eg als. We will wo k wi h he de ini ion in he la e pape , whe e,
in pa icula , he que maßin eg al W1(su ace a ea) o a log-conca e unc ion is
de ined by
W1( ) := Z∞
0
W1({x∈Rn: (x)≥ })d .
By C o on’s o mula (c . [32, p. 235]), his equals
W1( ) = cnZAn,1
max
z∈E (z)dµn,1(E),
whe e cn=|Bn
2|
|Bn−1
2|is a cons an depending only on nand An,1is he se o a ine
1-dimensional subspaces o Rnand µn,1is he Haa p obabili y measu e on i .

6 DAVID ALONSO, BERNARDO GONZ´
ALEZ, C. HUGO JIM´
ENEZ, RAFAEL VILLA
Theo em 2.3. Le , g :Rn→Rbe wo in eg able log-conca e unc ions wi h
ull-dimensional suppo and con inuous when es ic ed o hei suppo s. Then
(10) ZRn
⋆ g(x)dx ≤2n
nk k∞kgk∞
W1(g)RRn (x)dx +W1( )RRng(x)dx
2 maxx0∈RnW1( (·)g(x0−·)) .
Fu he mo e, when n≥3 his inequali y becomes an equali y i and only i (x)
k k∞=
g(−x)
kgk∞is he cha ac e is ic unc ion o an n-dimensional simplex.
Finally, we will p o e he ollowing ex ension o (7). Be o e s a ing i le us s a
wi h he ollowing de ini ion:
De ini ion 2.1. Le , g :Rn→Rbe wo in eg able log-conca e unc ions. De ine
⊕g(z) := sup
2z=x+yp (x)g(y) = p ⋆ g(2z).
Following he p oo gi en in [17] o inequali y (7), we can show he ollowing
esul . The inequali y in he esul was also ob ained in [3] in he mo e gene al case
whe e z=λx + (1 −λ)yand no jus λ=1
2. Howe e , equali y cases need a mo e
de ailed a gumen .
Theo em 2.4. Le , g :Rn→Rbe wo in eg able log-conca e unc ions wi h
ull-dimensional suppo s and con inuous when es ic ed o hem. Then
(11) ZRnp (x)¯g(x)dx ZRn
⊕g(x)dx ≤2nZRn
(x)dx ZRn
g(x)dx.
Equali y holds i and only i he ollowing wo condi ions a e sa is ied:
(i) supp =supp ¯gis a ansla ion o a cone Cwi h e ex a 0 wi h simplicial
sec ion, and
(ii) (x) = c1e−ha,xion supp and g(x) = c2e−hb,xion supp g o some c1, c2>0
and some a, b ∈Rnsuch ha ha, xi ≥ 0≥ hb, xi o e e y x∈C.
3. (θ, )-con olu ion bodies o log-conca e unc ions and
Roge s-Shepha d inequali ies
In his sec ion we p o e he a o emen ioned ex ensions o Roge s-Shepha d in-
equali y o log-conca e unc ions. In o de o p o e hem we need o in oduce
some mo e no a ion. Gi en , g wo in eg able log-conca e unc ions wi h ull-
dimensional suppo , x∈supp + supp gand ∈(0,1], le us ecall ha we
deno e
A (x) = A ( , g)(x) := {z∈supp ∩(x−supp g) : (z)g(x−z)≥ k k∞kgk∞}.
Since and ga e in eg able log-conca e unc ions, A (x) is a bounded con ex se .
De ini ion 3.1. Le , g :Rn→Rbe wo in eg able log-conca e unc ions wi h
ull-dimensional suppo , ∈(0,1],θ∈[0,1]. We de ine he (θ, )-con olu ion se
o and gas he se
Cθ, =Cθ, ( , g) := {x∈supp +supp g:A (x)6=∅,|A (x)| ≥ θM }
whe e
M =M ( , g) := max
x0∈supp +supp g|A (x0)|.
ROGERS-SHEPHARD INEQUALITY FOR LOG-CONCAVE FUNCTIONS 7
Rema k. When (x) = χK(x) and g(x) = χL(x) a e he cha ac e is ic unc ions o
wo con ex bodies Kand L, he se s A (x) = K∩(x−L) o any ∈(0,1] and we
eco e he de ini ion o he θ-con olu ion bodies K+θLin [2].
I is ob ious om he de ini ion ha o any ixed , he se s Cθ, dec ease on
θ. The ollowing lemma implies he con exi y o hese se s and gi es a e e se
(inc easing on θ) inclusion ela ion when no malized by he igh ac o , as shows
Co olla y 3.2.
Lemma 3.1. Le ∈(0,1], , g :Rn→Rbe wo in eg able log-conca e unc ions
wi h ull-dimensional suppo such ha M =|A (0)|,θ1, θ2, λ1, λ2∈[0,1] wi h
λ1+λ2≤1. Then
λ1Cθ1, +λ2Cθ2, ⊆ Cθ, ,
wi h 1−θ1
n=λ1(1 −θ
1
n
1) + λ2(1 −θ
1
n
2).
P oo . Le x1∈ Cθ1, ,x2∈ Cθ2, . Fo any z0∈ A (0), z1∈ A (x1), z2∈ A (x2), he
log-conca i y o and gimplies
((1 −λ1−λ2)z0+λ1z1+λ2z2)g(λ1x1+λ2x2−(1 −λ1−λ2)z0−λ1z1−λ2z2)
≥( (z0)g(−z0))1−λ1−λ2( (z1)g(x1−z1))λ1( (z2)g(x2−z2))λ2≥ k k∞kgk∞.
Thus,
A (λ1x1+λ2x2)⊇(1 −λ1−λ2)A (0) + λ1A (x1) + λ2A (x2)
and, by B unn-Minkowski inequali y
|A (λ1x1+λ2x2)|1
n≥
(1 −λ1−λ2)|A (0)|1
n+λ1|A (x1)|1
n+λ2|A (x2)|1
n≥
(1 −λ1−λ2)M
1
n
+λ1θ
1
n
1M
1
n
+λ2θ
1
n
2M
1
n
=
(1 −λ1(1 −θ
1
n
1)−λ2(1 −θ
1
n
2))M
1
n
.
Consequen ly, λ1x1+λ2x2∈ Cθ, .
In pa icula , aking θ1=θ2and λ1+λ2= 1 we ob ain ha hese se s Cθ, a e
con ex. Besides
Co olla y 3.2. Le ∈(0,1], , g :Rn→Rbe wo in eg able log-conca e unc ions
wi h ull-dimensional suppo such ha M =|A (0)|,0≤θ0≤θ < 1. Then
Cθ0,
1−θ
1
n
0⊆Cθ,
1−θ1
n
.
P oo . Taking θ1=θ2=θ0in he p e ious lemma, we ha e ha o any λ1, λ2∈
[0,1] wi h λ1+λ2≤1
(λ1+λ2)Cθ0, =λ1Cθ0, +λ2Cθ0, ⊆ Cθ, ,
wi h (λ1+λ2)(1 −θ
1
n
0) = 1 −θ1
n. Thus, o any θ0≤θ≤1, aking λ1+λ2=1−θ1
n
1−θ
1
n
0
we ob ain he esul . 
In a simila way we can p o e he ollowing:
8 DAVID ALONSO, BERNARDO GONZ´
ALEZ, C. HUGO JIM´
ENEZ, RAFAEL VILLA
Lemma 3.3. Le , g :Rn→Rbe wo in eg able log-conca e unc ions wi h ull-
dimensional suppo . Then o any 1, 2∈(0,1] and any λ∈[0,1] we ha e
M
1
n
λ
1 1−λ
2≥λM
1
n
1+ (1 −λ)M
1
n
2.
Consequen ly, M is con inuous on (0,1).
P oo . Le x1, x2be such ha M i=|A i(xi)| o i= 1,2. Since and ga e
log-conca e we ha e ha o any z1∈ A 1(x1), z2∈ A 2(x2) and any λ∈,[0,1]
(λz1+ (1 −λ)z2)g(λx1+ (1 −λ)x2−(λz1+ (1 −λ)z2))
≥( (z1)g(x1−z1))λ( (z2)g(x2−z2))1−λ
≥ λ
1 1−λ
2k k∞kgk∞.
Thus,
A λ
1 1−λ
2(λx1+ (1 −λ)x2)⊇λA 1(x1) + (1 −λ)A 2(x2).
By B unn-Minkowsi inequali y
M
1
n
λ
1 1−λ
2≥ |A λ
1 1−λ
2(λx1+ (1 −λ)x2)|1
n≥λM
1
n
1+ (1 −λ)M
1
n
2.
Consequen ly, he unc ion (s) = M
1
n
esis conca e in (−∞,0] and hen i is con in-
uous on (−∞,0). Thus, M = n(log ) is con inuous on (0,1). 
Le us now p o e inequali y (8).
P oo o Theo em 2.1 (inequali y). We can assume, wi hou loss o gene ali y, ha
k k∞=kgk∞= 1. By de ini ion o Cθ, , we ha e ha o any ∈(0,1]
C0, ={x∈supp + suppg:A (x)6=∅}
={x∈supp + suppg: ⋆ g(x)≥ }.
Fo any ∈(0,1], le x0( )∈Rnbe such ha M =|A (x0)|. By Co olla y 3.2
wi h θ0= 0 and g eplaced by g(·+x0), o any θ∈[0,1]
(1 −θ1
n)(−x0( ) + C0, )⊆ −x0( ) + Cθ, .
Taking olumes and in eg a ing in θ∈[0,1] we ob ain
|C0, | ≤ 2n
nZ1
0|Cθ, |dθ =2n
nZRn
|A (x)|
M
dx.
Consequen ly
M |C0, | ≤ 2n
nZRn|A (x)|dx
and, in eg a ing in ∈(0,1]
Z1
0
M |C0, |d ≤2n
nZ1
0ZRn|A (x)|dxd .
The in eg al on he igh -hand side is
Z1
0ZRn|A (x)|dxd =ZRnZRn
(z)g(x−z)dzdx =ZRn
(x)dx ZRn
g(x)dx.
ROGERS-SHEPHARD INEQUALITY FOR LOG-CONCAVE FUNCTIONS 9
On he o he hand, he in eg al on he le -hand side is
Z1
0
M |C0, |d =ZRnZ ⋆g(x)
0
max
x0∈Rn|A (x0)|d dx
≥max
x0∈RnZRnZ ⋆g(x)
0|A (x0)|d dx
= max
x0∈RnZRnZRn
min { ⋆ g(x), (z)g(x0−z)}dzdx.
Since k k∞=kgk∞= 1, bo h quan i ies in he minimum a e smalle han o equal
o 1, he minimum is bounded om below by he p oduc and so, his quan i y is
bounded om below by
max
x0∈Rn ∗g(x0)ZRn
⋆ g(x)dx.
Thus
k ∗gk∞ZRn
⋆ g(x)dx ≤2n
nk k∞kgk∞ZRn
(x)dx ZRn
g(x)dx.

Le us now conside he case in which g(x) = (−x). We will deno e his unc ion
¯
and A ( )(x) := A ( , ¯
)(x). No ice ha o any ∈(0,1)
A ( )(0) = z∈Rn: (z)2≥ k k2
∞
=nz∈Rn: (z)≥√ k k∞o.
Analogously, le us deno e M ( ) := M ( , ¯
) and Cθ, ( ) := Cθ, ( , ¯
).
The ollowing lemma shows ha he maximum alue o |A ( )(x)|is a ained a
x= 0.
Lemma 3.4. Le :Rn→Rbe an in eg able log-conca e unc ion wi h ull-
dimensional suppo , hen o any ∈(0,1),
A ( )(x)⊆1
2x+A ( )(0).
Consequen ly, M ( ) = |A ( )(0)|.
P oo . Since is log-conca e, o any x∈supp −supp
A ( )(x) = (z∈Rn:s (z)
k k∞
(z−x)
k k∞≥√ )
⊆z∈Rn: (z−1
2x)
k k∞≥√ 
=1
2x+A ( )(0).

The ollowing lemma shows a ela ion be ween he (θ, )-con olu ion bodies o
and ¯
and he θ-con olu ion bodies o A ( )(0) and −A ( )(0).
16 DAVID ALONSO, BERNARDO GONZ´
ALEZ, C. HUGO JIM´
ENEZ, RAFAEL VILLA
No ice ha aking θ ending o 1 in (i) we ha e ha o e e y ∈(0,1) he
maximum o |A (x)|is only a ained a x0( ). Besides, by he con inui y o M , he
con inui y o |A (x0)|in and in x0, (ii) holds i and only i x0( ) is he same o
e e y ∈(0,1), and hus we may suppose wi hou loss o gene ali y ha x0( ) = 0.
Thus
(i) o e e y θ, ∈(0,1), (1 −θ1
n)C0, =Cθ, ,
(ii) o e e y ∈(0,1), maxx0∈Rn|A (x0)|=|A (0)|
(iii) o e e y x, z ∈Rn, min{ ⋆ g(x), (z)g(−z)}= ⋆ g(x) (z)g(−z).
Fi s o all no ice ha i g(x) = χK(x) is he cha ac e is ic unc ion o a con ex
body, hen
A (x) = A 2( )(0) ∩(x−K)
and
Cθ, =A 2( )(0) +θK,
whe e A 2( )(0) +θKdeno es he θ-con olu ion o con ex bodies de ined in (5).
Thus, since he (θ, )-con olu ion bodies o he unc ions a e he θ-con olu ion
bodies o some con ex bodies we ha e equali y in (i) i and only i o e e y ∈(0,1)
A 2( )(0) = −Kis a simplex and, consequen ly (x) = g(−x) is he cha ac e is ic
unc ion o a simplex.
We will p o e ha in he equali y case necessa ily one o he unc ions is he
cha ac e is ic unc ion o a con ex body.
Condi ion (iii) occu s i and only i ⋆ g(x) o (z)g(−z) equals 0 o 1, o e e y
x, z ∈Rn. Fi s , assume ha ⋆ g(x) = 1 o e e y x∈supp + supp g. Then o
e e y x∈supp + supp g,A1( )(0) ∩(x−A1(g)(0)) 6=∅and so
A1( )(0) + A1(g)(0) = supp + supp g.
Consequen ly and ga e cha ac e is ic unc ions.
Le us now assume ha he e exis s x∈supp + supp gsuch ha ⋆ g(x)<1.
Then o e e y z∈Rn, (z)g(−z) equals 0,1 and hen, o e e y ∈(0,1], A (0) =
A1(0). In such case he unc ion M is cons an in (0,1]. In pa icula i is also
con inuous on = 1 and hen (i) also holds o = 1.
No ice ha i |A1(0)|=|A (0)|= 0, hen o e e y ∈(0,1) we ha e ha o
e e y θ∈(0,1)
Cθ, ={x∈supp +suppg :A (x)6=∅},
con adic ing (i). Thus, i we ha e equali y in (8), |A1(0)|>0.
Now, i (i) holds, we ix ∈(0,1] and ake x∈supp( ) + supp(g). I x /∈ C0,
hen A (x) = ∅. I x∈ C0, hen he e exis s θx∈[0,1] such ha x∈∂Cθx, and
x= (1 −θ
1
n
x)y o some y∈∂C0, . Thus, we ha e equali y in
θ
1
n
x|A (0)|1
n=|A (x)|1
n=|A (θ
1
n
x0 + (1 −θ
1
n
x)y)|1
n
≥θ
1
n
x|A (0)|1
n+ (1 −θ
1
n
x)|A (y)|1
n≥θ
1
n
x|A (0)|1
n.
and, by he equali y cases in B unn-Minkowski inequali y, A (x) is homo he ic o
A (0). Thus, o e e y x∈Rnand ∈(0,1], A (x) is ei he emp y o a homo he ic
copy o A (0). Mo eo e , i we pa icula ize in = 1, we ha e ha
A1(x) ={z: (z)g(x−z) = 1}={z: (z) = 1}∩(x+{z:g(−z) = 1}) and
A1(0) ={z: (z)g(−z) = 1}={z: (z) = 1}∩{z:g(−z) = 1}

ROGERS-SHEPHARD INEQUALITY FOR LOG-CONCAVE FUNCTIONS 17
and o any θ∈[0,1] hen Cθ,1is he θ-con olu ion o he con ex bodies
Cθ,1=A1( )(0) +θA1(g)(0).
Thus, by P oposi ion 2.10 in [2], he con ex bodies A1( )(0) and −A1(g)(0) a e
he same simplex A1(0).
Le us now assume ha none o he unc ions , g is a cha ac e is ic unc ion,
and we will ind a con adic ion.
Since and ga e no cha ac e is ic unc ions he e exis 0 < 1, 2<1 and
z1, z2∈Rnsuch ha 1≤ (z1)<1 and 2≤g(−z2)<1. Le us deno e by F1a
ace o A1(0), wi h ou e no mal ec o u1and con ained in he hype plane {x∈
Rn:hx, u1i=c1}, such ha A1(0) ⊆ {x∈Rn:hx, u1i ≤ c1}and hz1, u1i> c1.
Analogously, le F2be a ace o A1(0), wi h ou e no mal ec o u2and con ained in
he hype plane {x∈Rn:hx, u2i=c2}, such ha A1(0) ⊆ {x∈Rn:hx, u2i ≤ c2}
and hz1, u2i> c2.
Obse e ha he log-conca i y o and gimply ha con {z1,A1(0)} ⊂ A 2
1( )(0)
and con {z2,A1(0)} ⊂ A 2
2(¯g)(0).
I F1=F2, hen
(con {z1,A1(0)}∩con {z2,A1(0)}) A1(0) 6=∅.
Le z0be a poin in his in e sec ion. Then z0∈ A 1 2(0) since
1 2≤ (z0)g(−z0)<1.
Howe e , his is no possible, since A 1 2(0) = A1(0).
I F16=F2, le xbe a ec o wi h a small enough no m and pa allel o he only
edge con ained in all n−1 ace s F3,...,Fn+1 o A1(0) di e en om F1and F2
and poin ing om F2 o F1such ha z1/∈x+A1(0) and A1(0) ∩(x+F2)6=∅,
(con {z1,A1(0)} {z1,A1(0)})∩(x+F1)6=∅,
and
A1(0) ∩(x+ con {z2,A1(0)} {z2,A1(0)})6=∅.
Obse e ha o any poin zin he i s in e sec ion, i holds 1 2< 1≤ (z)g(x−
z), whe eas i z′is on he second, hen 1 2< 2≤ (z′)g(x−z′) and, since xpoin s
om F2 o F1,z′does no belong o x+A1(0).
Besides, o any o he ace Fi∩(x+Fi)6=∅and o any poin z′′ in Fi∩(x+Fi)
we ha e (z′′)g(x−z′′) = 1. These h ee ypes o poin s belong o he se A 0 1(x),
which we ha e p o ed ha is a homo he ic copy o he simplex A 1 2(0) = A1(0).
Then, since o e e y ace Fi he e exis poin s in (x+Fi)∩A 1 2(x) we ha e ha
x+A1(0) ⊆A 1 2(x) and since he poin s z′∈A 1 2(x) (x+A1(0)) he inclusion
is s ic . Thus
|A 1 2(0)|=|A1(0)|=|x+A1(0)|<|A 1 2(x)|,
con adic ing (ii).
Thus, o gmus be a cha ac e is ic unc ion and, consequen ly, (x) = g(−x)
is he cha ac e is ic unc ion o a simplex. 
The p oo o he equali y cases in (10) ollows he same lines.
P oo o Theo em 2.3 (Equali y). Wi hou loss o gene ali y we assume ha || ||∞=
||g||∞= 1.
18 DAVID ALONSO, BERNARDO GONZ´
ALEZ, C. HUGO JIM´
ENEZ, RAFAEL VILLA
Then, equali y holds in (10) i and only i i holds equali y on each inequali y all
along he p oo o (10). Mo e pa icula ly, i x0( ) is such ha M1, =W1(A (x0))
we ha e ha
(i) o e e y θ, ∈(0,1), (1 −θ1
n−1)(−x0( ) + Cn−1
0, ) = −x0( ) + Cn−1
θ, ,
(ii)
max
x0∈RnZRnZ ⋆g(x)
0
W1(A (x0))d dx =ZRnZ ⋆g(x)
0
max
x0∈RnW1(A (x0( )))d dx
(iii) o e e y x∈Rn, E ∈An,1,
min{ ⋆ g(x),max
z∈E (z)g(x0( )−z)}= ⋆ g(x) max
z∈E (z)g(x0( )−z)
(i ) o e e y θ∈Sn−1and o e e y z, w ∈θ⊥, , s ∈R,
max
s∈R z(s)gw( −s) = max
s∈Rmin{ z(s)kgwk, gw( −s)k zk∞}.
Taking θ ending o 1 in (i) we ha e ha o e e y ∈(0,1] he maximum o
W1(A (x)) is only a ained a x0( ). Besides, (ii) holds i and only i x0( ) is he
same o e e y , and hus we may suppose wi hou loss o gene ali y ha x0( ) = 0.
Thus
(i) o e e y θ, ∈(0,1), (1 −θ1
n−1)Cn−1
0, =Cn−1
θ, ,
(ii) o e e y ∈(0,1], maxx0∈RnW1(A (x0)) = W1(A (0))
(iii) o e e y x∈Rn, E ∈An,1,
min{ ⋆ g(x),max
z∈E (z)g(−z)}= ⋆ g(x) max
z∈E (z)g(−z),
(i ) o e e y θ∈Sn−1and o e e y z, w ∈θ⊥, , s ∈R,
max
s∈R z(s)gw( −s) = max
s∈Rmin{ z(s)kgwk, gw( −s)k zk∞}.
As in he p e ious case, we ha e ha i g(x) = χK(x) is he cha ac e is ic
unc ion o a con ex body, hen he (n−1)- h (θ, )-con olu ion bodies o he
unc ions a e he (n−1)- h θ-con olu ion bodies o some con ex bodies and, as i
was p o ed in [1], i n≥3 we ha e equali y in (i) i and only i o e e y ∈(0,1)
A 2( )(0) = −Kis a simplex and, consequen ly (x) = g(−x) is he cha ac e is ic
unc ion o a simplex.
We will p o e ha in he equali y case necessa ily one o he unc ions is he
cha ac e is ic unc ion o a con ex body.
Condi ion (iii) occu s i and only i ⋆ g(x) o maxz∈E (z)g(−z) equals 0 o 1,
o e e y x∈Rn, E ∈An,1. As we ha e seen in he p e ious case, i ⋆ g(x) = 1
o e e y x∈supp + supp g hen and ga e cha ac e is ic unc ions.
Le us now assume ha he e exis s x∈supp + supp gsuch ha ⋆ g(x)<1.
Then o e e y E∈An,1maxz∈E (z)g(−z) equals 0 o 1. Consequen ly, o e e y
∈(0,1] A (0) = A1(0) because o he wise he e exis s some ∈(0,1) and some
z∈Rnsuch ha ≤ (z)g(−z)<1 and since z /∈ A1(0) he e exis 1-dimensional
a ine subspaces passing h ough zand no in e sec ing A1(0) and o all such
subspaces we would ha e ≤maxz∈E (z)g(−z)<1. Like be o e, in such case (i)
also holds o = 1.
No ice ha i W1(A1(0)) = W1(A (0)) = 0, hen o e e y ∈(0,1) we ha e
ha o e e y θ∈(0,1)
Cθ, ={x∈supp +suppg :A (x)6=∅},
ROGERS-SHEPHARD INEQUALITY FOR LOG-CONCAVE FUNCTIONS 19
con adic ing (i). Thus, i we ha e equali y in (8), W1(A1(0)) >0.
Now, i (i) holds, we ix ∈(0,1] and ake x∈supp( ) + supp(g). I x /∈ C0,
hen A (x) = ∅. I x∈ C0, hen he e exis s θx∈[0,1] such ha x∈∂Cθx, and
x= (1 −θ
1
n
x)y o some y∈∂C0, . Thus, we ha e equali y in
θ
1
n
xW1(A (0)) 1
n−1=W1(A (x)) 1
n−1=W1(A (θ
1
n
x0 + (1 −θ
1
n
x)y)) 1
n−1
≥θ
1
n
xW1(A (0)) 1
n−1+ (1 −θ
1
n
x)W1(A (y)) 1
n−1
≥θ
1
n
xW1(A (0)) 1
n−1,
and, by he equali y cases in B unn-Minkowski inequali y o que maßin eg als, i
n≥3 hen A (x) is homo he ic o A (0). Thus, o e e y x∈Rnand ∈(0,1],
A (x) is ei he emp y o a homo he ic copy o A (0). Pa icula izing a = 1, we
ha e ha o any θ∈[0,1] Cn−1
θ,1is he (n−1)- h θ-con olu ion o he con ex bodies
Cn−1
θ,1=A1( )(0) +n−1,θ A1(g)(0).
Thus, i (i) holds hen by he cha ac e iza ion o he equali y cases in [1] A1( )(0)
and −A1(g)(0) a e he same simplex A1(0).
Now, i we assume ha nei he o he unc ions , g is a cha ac e is ic unc ion,
wi h he same p oo as be o e we ind a con adic ion. Thus, o gmus be
a cha ac e is ic unc ion and, consequen ly, (x) = g(−x) is he cha ac e is ic
unc ion o a simplex. 
6. Colesan i’s inequali y o wo unc ions
In his sec ion we show ha Colesan i’s unc ional e sion o Roge s-Shepha d
inequali y (7) can be ex ended o he case in which we conside any pai o unc ions
and no necessa ily g(x) = ¯
(x). Le us ecall ha simila esul s we e ob ained in
[3].
P oo o Theo em 2.4. Fo any z∈Rnsuch ha ⊕g(z)>0 le xz, yz∈Rnbe
such ha ⊕g(z) = p (xz)g(yz) wi h 2z=xz+yz. No ice ha xzand yzexis
by Rema k 2, since ⊕g(z) = p ⋆ g(2z). Using he log-conca i y o and g,
(12) (x)g(z−x)≥p (xz)g(yz)p (2x−xz)g(xz−2x)
o e e y x∈Rn. In eg a ing in x∈Rn
∗g(z)≥p (xz)g(yz)ZRnp (2x−xz)¯g(2x−xz)dx
=1
2n ⊕g(z)ZRnp (x)¯g(x)dx.
In eg a ing in z∈Rnwe inally ob ain
ZRn
(x)dx ZRn
g(x)dx ≥1
2nZRn
⊕g(z)dz ZRnp (x)¯g(x)dx,
as wan ed.
Le us now cha ac e ize he equali y cases. I (i) and (ii) a e sa is ied, hen he e
exis s p∈Rnsuch ha
supp = supp ¯g=p+C
20 DAVID ALONSO, BERNARDO GONZ´
ALEZ, C. HUGO JIM´
ENEZ, RAFAEL VILLA
and o e e y z∈supp + supp gwe ha e ha
supp ∩(z−supp g) = (p+C)∩(z+p+C) = p+C∩(z+C)
and since Chas a simplicial sec ion, his equals
supp ∩(z−supp g) = p′+C
o some p′∈Rn. We will show ha p′=p
2+xz
2 o xzsuch ha ⊕g(z) =
p (xz)g(2z−xz). No ice ha , as be o e, such xzexis s, because o he con inui y
p ope ies o and gon hei suppo s. In such case we would ha e ha o e e y
z∈supp + supp g
supp ∩(z−supp g) = xz
2+1
2(supp ∩supp ¯g)
and hen o e e y z∈supp + supp gand e e y x∈supp ∩(z−supp g)
•2x−xz∈supp , and
•xz−2x∈supp g.
Then, inequali y (12) holds wi h equali y o he unc ions (x) = c1e−ha,xi,x∈
supp and g(x) = c2e−hb,xi,x∈supp g o e e y x, z ∈Rn.
In o de o show ha o e e y z∈supp + supp gwe ha e p′=p
2+xz
2, no ice
ha
2(supp + supp g) = 2C−2C=C−C= supp + supp g
and so, 2z∈supp + supp gand o e e y x∈supp ∩(2z−supp g)
p (x)g(2z−x) = √c1c2e−ha, x
2ie−hb,z−x
2i
=√c1c2e−hb,zie−ha−b, x
2i
and so,
⊕g(z) = √c1c2e−hb,zie−min{ha−b, x
2i:x∈supp ∩(2z−supp g)}
=√c1c2e−hb,zie−min{ha−b,¯xi:¯x∈p
2+C∩(z+C)}.
Since (C∩z+C) = −p+ supp ∩(z−supp g) = −p+p′+Cwe ha e ha
min
¯x∈p
2+(C∩z+C)ha−b, ¯xi= min
¯x∈− p
2+p′+Cha−b, ¯xi
and since ha−b, xi ≥ 0 o e e y x∈C, he minimum is a ained when ¯x=xz
2=
−p
2+p′. Thus p′=p
2+xz
2.
Le us now p o e ha (i) and (ii) a e necessa y condi ions o equali y in (11)
o hold. We can assume, wi hou loss o gene ali y, ha (x0) = k k∞=kgk∞=
g(y0) = 1 and le us w i e (x) = e−u(x)and g(x) = e− (x) o some con ex
unc ions u, . No ice ha , since k k∞=kgk∞= 1, uand ake alues in
[0,+∞].
Equali y in (11) happens i and only i o e e y z∈supp +supp gand e e y x∈
supp ∩(z−supp g) we ha e equali y in (12). Thus, o e e y z∈supp +supp g
he suppo o bo h unc ions as unc ions o x∈Rnmus be he same and so
supp ∩(z−supp g) = xz
2+1
2(supp ∩(−supp g)),
whe e xzis such ha ⊕g(z) = p (xz)g(2z−xz). No ice ha in pa icula his
implies ha supp ∩(−supp g) is ull-dimensional, since we a e assuming ha
supp and supp ga e ull-dimensional and hen he e exis s some z∈supp +
ROGERS-SHEPHARD INEQUALITY FOR LOG-CONCAVE FUNCTIONS 21
supp gsuch ha supp ∩(z−supp g) is ull-dimensional and hen (supp ∩
(−supp g)) is ull-dimensional.
Besides, o e e y z∈supp + supp gwe ha e equali y in (12) o e e y x∈
supp ∩(z−supp g) i and only i
u1
2(2x−xz) + xz
2=1
2(u(2x−xz) + u(xz))
1
2(xz−2x) + yz
2=1
2( (xz−2x) + (yz))
o e e y x∈supp ∩(z−supp g), whe e xz, yz∈Rna e such ha ⊕g(z) =
p (xz)g(yz). In pa icula , o z0=x0+y0
2we ha e ha ⊕g(z0) = p (x0)g(y0)
and so
u1
2(2x−x0) + x0
2=1
2u(2x−x0)
1
2(x0−2x) + y0
2=1
2 (x0−2x).
Consequen ly, o e e y x′=x−x0and y′=x0
2−y0
2−x
u(x0+x′) = 1
2u(x0+ 2x′)
(y0+y′) = 1
2 (y0+ 2y′).
Thus, supp ∩(z0−supp g) = x0
2+1
2(supp ∩(−supp g)) is a closed con ex cone
x0+C(wi h Ca cone wi h e ex a 0) and so
supp ∩(−supp g) = x0+C
and o e e y z∈supp + supp g
supp ∩(z−supp g) = xz
2+x0
2+C.
Fu he mo e, supp g∩(z0−supp ) = z0−supp ∩(z0−supp g) = z0−x0−C
is a closed con ex cone y0+C′. This implies ha C′=−C. Besides, i he e ex
o he cone is unique, z0−x0=y0and hen y0=−x0which implies z0= 0. I
he e ex is no unique hen, since bo h z0−x0and y0a e e ices o he cone
y0−C, also y0+ 2(z0−x0−y0) = y0−2z0=−x0and so y0−C=−x0−Cand
−x0+C=y0+C.
On he o he hand, o any z∈supp + supp gi x∈supp ∩(z−supp g),
hen
2x−xz∈x0+C.
I equali y holds in (12) hen uis a ine in any segmen ha connec s x0+Cwi h
a poin xzand is a ine in any segmen ha connec s −x0−Cand some yz.
Le us now ake z∈x0+ supp g. Then −x0+z∈supp gand, since x0∈supp
we ha e ha −x0+z∈(z−supp )∩supp g=z−supp ∩(z−supp g) =
z−xz
2−x0
2−Cand so xz
2∈x0
2−Cand
xz∈x0−C.
Consequen ly xz=x0. O he wise, conside he ay om xz ha passes h ough x0.
Since xz∈x0−Cany poin pin his ay such ha he segmen [xz, p] con ains x0

22 DAVID ALONSO, BERNARDO GONZ´
ALEZ, C. HUGO JIM´
ENEZ, RAFAEL VILLA
is con ained in x0+Cand since u≥0 is a ine in he segmen [xz, p] and u(x0) = 0
hen u= 0 o any such p, con adic ing he in eg abili y o .
Thus, o any z∈x0+ supp gwe ha e ha xz=x0and yz= 2z−x0. No ice
ha hen o e e y z∈x0+ supp gwe ha e ha
supp ∩(z−supp g) = xz
2+x0
2+C=x0+C
as xz=x0. Consequen ly, supp = supp ∩(−supp g) since o e e y y∈supp ,
as Cis a ull-dimensional cone, we can ake w∈Csuch ha y∈ −w+ (x0+C).
Thus, i we ake z=−w∈ −C=x0−(x0+C)⊂x0+ supp g, we ha e ha
y∈z+(x0+C)⊂z−supp gand so y∈supp ∩(z−supp g) = supp ∩(−supp g).
Consequen ly supp =x0+C⊆ −supp g.
Analogously, ake z∈ −x0+ supp . Since −x0∈supp g, we ha e ha x0+z∈
supp ∩(z−supp g) = xz
2+x0
2+Cand, consequen ly yz
2∈ −x0
2+C. Thus
yz∈ −x0+C=y0+C. Consequen ly yz=y0. O he wise, conside he ay om
yz ha passes h ough y0. Since yz∈y0+Cany poin pin his ay such ha
he segmen [yz, p] con ains y0is con ained in y0−Cand since ≥0 is a ine in
he segmen [yz, p] and (x0) = 0 hen = 0 o any such p, con adic ing he
in eg abili y o g.
Thus, o any z∈ −x0+ supp we ha e ha xz= 2z−y0and yz=y0. No ice
ha hen o e e y z∈ −x0+ supp we ha e ha
supp ∩(z−supp g) = xz
2+x0
2+C=z−y0
2+x0
2+C=z+x0+C,
since x0+C=−y0+C. Consequen ly, −supp g= supp ∩(−supp g) since o
e e y y∈ −supp g, as Cis a ull-dimensional cone, we can ake w∈Csuch ha
y∈ −w+ (x0+C). Thus, i we ake z=w∈C=−x0+ supp gwe ha e ha
y+z∈supp ∩(z−supp g) = z+x0+Cand so y∈x0+C= supp .
Consequen ly −supp g⊆supp and so −supp g= supp =x0+C.
Now le us see ha is a ine on −x0−C. Le us ake x, y ∈ −x0−Cand
conside z=y
2+x0
2∈ −C=x0+ supp g. Then, o his z,yz=yand since
x∈ −x0−C, is a ine in he segmen ha connec s xand y. Consequen ly, is
a ine on −x0−Cand so g(x) = c2e−hb,xion −x0−C. Thus, Cdoes no con ain
any s aigh line lsince o he wise, as is a ine and posi i e, i mus be cons an
on he line land so Chas only one e ex and x0=−y0and hb, xi<0 o e e y
x∈C {0}.
Analogously, uis also a ine on x0+Csince o any x, y ∈x0+C, i we ake
z=x
2−x0
2∈C=−x0+ supp we ha e ha o his z,xz=x−x0−y0=xand
so, uis a ine on x0+Cand (x) = c1e−ha,xion x0+C. Since k k∞= (x0) we
ha e ha ha, xi>0 o e e y x∈C.
Finally, conside ing he sec ion o Cby a hype plane, since he in e sec ion o
his sec ion wi h any o i s ansla es is homo he ic o i sel , he sec ion mus be a
simplex. 
Rema k. Theo em 2.4 becomes inequali y (7) i g(x) = ¯
(x) because ⊕g(z) =
∆ (z). Mo eo e , i also eco e s (4) when we pa icula ize (x) = e−hK(x), g(x) =
e−hL(−x), whe e hKand hLa e he suppo unc ions o wo con ex bodies Kand
L ha con ain he o igin. Then ⊕g(x) = e−hK∩L(x)and p (x)g(x) = e−hK−L
2
(x),
ROGERS-SHEPHARD INEQUALITY FOR LOG-CONCAVE FUNCTIONS 23
and since ZRn
e−hK(x)dx =n!|K◦|,
hen (11) becomes
|(K∩L)◦|K−L
2◦≤2n|K◦||L◦|.
F om he cha ac e iza ion o he equali y cases in (11), his only con e ges o equal-
i y when we conside sequences o se s (K◦
n)nand (−L◦
n)ncon e ging o simplices
wi h 0 in one o he e ices and he same ou e no mal ec o s a he ace s ha
pass h ough he o igin.
Taking in o accoun ha (K∩L)◦= con {K◦, L◦}and K−L
2⊂con {K, −L},
i we change he ole o K◦and L◦by Kand −L o simplici y, hen
|K∩L||con {K, −L}| ≤ K◦−L◦
2◦|con {K, −L}| ≤ 2n|K||L|,
showing he asse ion and sligh ly s eng hening (4). In o de o ha e equali y in (4)
we mus ha e Kand Lbe simplices wi h 0 in one o he e ices and he same ou e
no mal ec o s a he ace s ha pass h ough he o igin and K+L
2= con {K, L}.
Thus K=Lis a simplex. O he wise he e exis s a di ec ion θ∈Sn−1such ha
hK(θ)< hL(θ) (o hL(θ)< hK(θ)). Thus, hK+L
2(θ)< hL(θ)≤hcon {K,L}(θ) (o
hK+L
2(θ)< hK(θ)≤hcon {K,L}(θ)).
ACKNOWLEDGEMENTS
Pa o his wo k was ca ied ou a he ‘Ins i u o de Ma em´a icas de la Uni e -
sidad de Se illa’ (IMUS) whe e he D. Alonso was in i ed and B. Gonz´alez ul illed
he p og am ‘Ayudas pa a es ancias co as pos doc o ales en el IMUS’, and hey
a e hank ul o he in i a ion and o he good wo king condi ions and en i onmen
he e.
D. Alonso is pa ially suppo ed by ‘Ins i u Uni e si a i de Ma em`a iques i Apli-
cacions de Cas ell´o’, Spanish Minis y o Sciences and Inno a ion (MICINN) p ojec
MTM2013-42105-P and BANCAJA p ojec P1-1B2014-35. B. Gonz´alez is pa -
ially suppo ed by Spanish Minis y o Economy and Compe i i eness (MINECO)
p ojec MTM2012-34037.
C. H. Jim´enez and R. Villa a e suppo ed by MINECO p ojec MTM2012-30748
and C. H. Jim´enez is also suppo ed by Capes and IMPA.
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E-mail add ess:alonsod@uniza .es
E-mail add ess:bg.me[email p o ec ed]e
E-mail add ess:hugojimenez@ma .puc- io.b
E-mail add ess: [email protected]
Uni e sidad de Za agoza
Technische Uni e si ¨
a M¨
unchen
Pon i ´
ıcia Uni e sidade Ca ´
olica do Rio de Janei o
Uni e sidad de Se illa