App oaching Eldan’s and Lee & Vempala’s bounds o he KLS
conjec u e in a uni ied me hod
Da id Alonso-Gu ié ez, Jesús Bas e o
Depa amen o de Ma emá icas - I.U.M.A.
Uni e sidad de Za agoza
Campus Plaza San F ancisco
50009 Za agoza, Spain
alonso[email p o ec ed], [email p o ec ed]
Re . Real Academia de Ciencias. Za agoza. 75: 85–110, (2020). ISSN: 0370-3207
Resumen
La p incipal idea de es e a ículo es e isa las p uebas de las mejo es es imaciones
conocidas pa a la conje u a KLS de sal o espec al, demos adas po Eldan y Lee &
Vempal, aplicando el esquema de localización de Eldan a dos sis emas de ecuaciones
di e enciales es ocás icas di e en es. Damos una p ueba uni icada de es as dos aco a-
ciones ob eniendo la es imación de Eldan desde el sis ema de ecuaciones di e enciales
es ocás icas conside ado po Lee & Vempala.
Abs ac
The main idea o his pape is o e iew he p oo on he bes known es ima es o
he KLS spec al gap conjec u e, gi en by Eldan and Lee & Vempala by applying El-
dan’s localiza ion scheme o wo di e en sys ems o s ochas ic di e en ial equa ions.
We gi e a uni ied p oo o hese wo bes bounds ob aining Eldan’s es ima e om
he sys em o s ochas ic equa ions conside ed by Lee & Vempala.
1 In oduc ion
The Kannan-Lo ász-Simono i s spec al gap conjec u e (KLS) is a majo p oblem in
asymp o ic geome ic analysis. I s o igin comes om heo e ical compu e sciences as a
p oblem a ising in he s udy o he complexi y o an sampling algo i hm and i is ela ed o
85
many o he b anches o ma hema ics like con ex geome y, p obabili y, PDE’s, Riemannian
geome y and in o ma ion o lea ning heo y (see [AB1], [BGVV] and he e e ences he ein
(o [AB2] o a p esen a ion o he conjec u e w i en in Spanish)).
I conce ns log-conca e p obabili ies and can be s a ed in he ollowing way:
Conjec u e 1.1 (KLS spec al gap conjec u e).The e exis s an absolu e cons an C > 0
such ha , o any log-conca e p obabili y µin Rn
(1) µ+(A)≥C
pkCo µkop
min{µ(A), µ(Ac)}, o any Bo el se A⊂Rn
whe e
µ+(A) = lim in
ε→0
µ(Aε)−µ(A)
ε,
being Aε={a+x:a∈A, |x|< ε}, and kCo µkop is he ope a o no m o he co a iance
ma ix o µ
This is a Cheege ype isope ime ic inequali y. This conjec u e was posed in [KLS],
whe e he au ho s p o ed he Cheege ype isope ime ic inequali y wi h cons an C
Eµ|x|
(whe e Eµ|x|deno es he expec ed alue o he Euclidean no m wi h espec o he p oba-
bili y µ) ins ead o C
√kCo µkop
.
The KLS conjec u e has an equi alen exp ession as a co esponding Poinca é ype
inequali y: he e exis s an absolu e cons an C > 0such ha
(2) ZRn| −Eµ |2dµ ≤CkCo µkop ZRn|∇ |2dµ
o any log-conca e p obabili y µin Rnand Lipschi z µ-in eg able unc ion .
The ac o kCo µkop appea ing in bo h exp essions (1) and (2) is jus a no maliza ion
ac o . Indeed, since he conjec u e in ol es e e y Bo el se A⊆Rn, o e e y Lipschi z
µ-in eg able unc ion in i s equi alen o m, making a change o a iables, we can assume
ha µis cen e ed and ha Co µ=In(iden i y ma ix), i.e. he new log-conca e measu e
is iso opic and hen we can e o mula e bo h conjec u es in he ollowing way: he e exis s
an absolu e cons an Csuch ha o any iso opic log-conca e p obabili y in Rn
(3) µ+(A)≥Cmin{µ(A), µ(Ac)}, o any Bo el se A⊂Rn
86
o , equi alen ly, he e exis s an absolu e cons an C
(4) ZRn| −Eµ |2dµ ≤CZRn|∇ |2dµ
o any iso opic log-conca e p obabili y µin Rnand any Lipschi z µ-in eg able unc ion
.
This conjec u e emains open and he bes es ima es known up o now, which depend
on he dimension, o he alue o he cons an in (1) and (2) ha e been ob ained in wo
di e en pape s by Eldan ([E1], see also [E2] o ano he app oach) and Lee & Vempala
([LV1], see also [LV2] o a nice su ey on his conjec u e), espec i ely.
The esul s whose p oo s we wan o uni y a e gi en by he ollowing wo heo ems:
Theo em 1.1 (Eldan, [E1]).The e exis s an absolu e cons an C > 0such ha o any
iso opic log-conca e p obabili y µin Rn
(5) µ+(A)≥C
σnlog nmin{µ(A), µ(Ac)} o any Bo el se A⊂Rn
whe e σn=qsup Eµ|X|−√n
2and he sup uns o e all iso opic log-conca e andom
ec o s Xin Rn.
Theo em 1.2 (Lee & Vempala, [LV1]).The e exis s an absolu e cons an C > 0such ha
o any iso opic log-conca e p obabili y µin Rn
(6) µ+(A)≥C
n1/4min{µ(A), µ(Ac)} o any Bo el se A⊂Rn.
The pa ame e σnappea ing in Eldan’s esul is ela ed wi h a di e en conjec u e,
which is he hin shell wid h conjec u e p oposed by Bobko -Koldobsky ([BK]): he e exis s
an absolu e cons an C > 0such ha o any iso opic, log-conca e p obabili y in Rnwe
ha e σµ=qEµ|x|−√n
2≤C.
I his conjec u e we e ue i would imply ha he mass in he iso opic log-conca e
p obabili ies is concen a ed in a hin shell a ound a dis ance √n om he o igin, Besides,
he esul (5) would imply ha he KLS conjec u e is ue up o a log n ac o . As i is also
e y well known ha he KLS conjec u e is s onge han he hin shell wid h conjec u e,
he esul (6) implies he bes known es ima e o he he las conjec u e, i.e. σn≤Cn1/4.
87
P e ious es ima es o his pa ame e we e ound by Kla ag [K] and Guedon-Milman [GM].
Mo e in o ma ion on hese ela ions can be seen, o ins ance, in [BGVV] and [AB1].
The p oo o bo h esul s, Theo ems 1.1 and 1.2, ollow he o iginal idea de eloped by
Eldan, he localiza ion scheme in oduced in [E1]: gi en an iso opic log-conca e p obabili y
µin Rn, a s ochas ic sys em o di e en ial equa ions o igina es a s ochas ic p ocess o (no
necessa ily iso opic) log-conca e p obabili ies (µ ) ≥0which a e an I ô p ocess. We can
ge “good” in o ma ion om some µTand hen come back o he o iginal µ. Howe e
he wo p oo s p opose di e en s ochas ic sys ems o di e en ial equa ions in o de o ge
s ochas ic p ocess (µ ) ≥0 om which we can ob ain es ima es.
The main pu pose o his pape is o uni y he wo app oaches and gi e a p oo o bo h
esul s oge he , which will ollow om he same s ochas ic sys em o di e en ial equa ions.
E en hough we a e no in oducing any uly new ideas in his pape , a he han ca e ully
mixing and gluing he a gumen s om he a o emen ioned au ho s, i is ou desi e o cla i y
and shed ligh on he a gumen s o his beau i ul and in e es ing heo y wha has mo ed
us o w i e his wo k and b ing i close o he in e es ed people e en i hey a e less expe
in he ield.
The heo em we a e going o p o e in his wo k is he ollowing, which collec s bo h
Theo ems 1.1 and 1.2.
Theo em 1.3. The e exis s an absolu e cons an C > 0such ha o any iso opic log-
conca e p obabili y µin Rn he ollowing isope ime ic inequali y holds
µ+(A)≥C
min{σnlog n, n1/4}min{µ(A), µ(Ac)}
o any Bo el se A⊆Rn.
The pape is o ganized in he ollowing way. In Sec ion 2 we will in oduce no a ion,
some de ini ions and some p e ious esul s we a e going o use in o de o de elop ou
p oo . In Sec ion 3 we will in oduce Eldan’s localiza ion scheme, p esen ing he sys em
o s ochas ic di e en ial equa ions we will conside in his wo k, which will de ine he
a o emen ioned s ochas ic p ocess o log-conca e p obabili ies (µ ) ≥0. In Sec ion 4 we will
gi e an o e iew o he s a egy we ollow in o de o s ess ou he e en whose p obabili y
is needed so ha he es ima es o he KLS cons an can be ob ained. The ace o he
co a iance ma ix o he p obabili ies (µ ) ≥0will be needed o bound he p obabili y o
88
such e en om below. They will be p o ed o be small enough wi h some p obabili y in
Sec ion 5. Finally, in Sec ion 6 we will pu all he inequali ies oge he o comple e he
p oo o Theo em 1.3.
2 No a ion and de ini ions
In his sec ion we will in oduce some no a ion and de ini ions which a e common in his
amewo k. Some well-known esul s will also be explained ei he by gi ing hei p oo o
a e e ence o i .
We will deno e by |·| he Euclidean no m in Rnand also he absolu e alue on R.Sn−1
will deno e he Euclidean uni sphe e. A p obabili y measu e µon Rnis called log-conca e
i o any compac subse s A, B ⊆Rnand o any 0≤λ≤1
µ((1 −λ)A+λB)≥µ(A)1−λµ(B)λ.
The ollowing heo em by Bo ell [B], cha ac e izes his kind o p obabili ies: Le µbe
a non degene a e log-conca e p obabili y measu e on Rn, (i.e. no concen a ed in any
hype plane). Then, µis log-conca e i and only i µis absolu ely con inuous wi h espec
o he Lebesgue measu e and i s densi y is log-conca e, i.e. dµ(x) = (x)dx =e−V(x)dx,
whe e he unc ion V:Rn→(−∞,∞]is con ex.
In he sequel we will use he p obabilis ic no a ion Eµg:= RRng(x)dµ(x)and Va µg:=
Eµ(g−Eµg)2 o any µ-in eg able unc ion g.
Fi s educ ions:
We say ha µis iso opic i i s ba ycen e bµ:= Eµx= 0 and i s co a iance ma ix
Co µ=Aµ:= Eµ(x−bµ)⊗(x−bµ) = In,
whe e Inis he iden i y ma ix. E e y non degene a e log-conca e p obabili y dµ(x) =
(x)dx admi s an a ine ans o ma ion such ha dν(y) = |de (Aµ)|1/2 (bµ+A1/2
µy)dy is
an iso opic log-conca e p obabili y. In pa icula wi h his change o a iables, i is easy
o p o e ha i an iso opic p obabili y µsa is ies Poinca é’s inequali y (2) wi h some
cons an C, hen o any non-degene a e linea map T he log-conca e p obabili y measu e
89

µ◦T, gi en by (µ◦T)(A) = µ(T(A)) o any Bo el se A, sa is ies (2) wi h he same
cons an C. The e o e, i he e exis s a cons an Cnsuch ha e e y iso opic log-conca e
p obabili y in Rnsa is ies (2) wi h cons an Cn, hen e e y log-conca e p obabili y in Rn
sa is ies (2) wi h he same cons an Cnand i he e exis s a cons an ˜
Cnsuch ha e e y
iso opic log-conca e p obabili y in Rnwi h compac suppo sa is ies (2) wi h cons an
˜
Cn, hen e e y log-conca e p obabili y in Rnwi h compac suppo sa is ies (2) wi h he
same cons an ˜
Cn.
Fu he mo e, i any log-conca e p obabili y in Rnwi h compac suppo sa is ies (2)
wi h some cons an Cn>2√2(which we can always assume), hen any iso opic log-
conca e p obabili y sa is ies (2) wi h cons an 5Cnand so any log-conca e (non-necessa ily
iso opic) p obabili y in Rnsa is ies (2) wi h cons an 5Cn. Indeed, le µbe an iso opic
log-conca e p obabili y, dµ =e−V(x)dx, wi h V:Rn→(−∞,∞]con ex and le be any
Lipschi z µin eg able unc ion . I we ake Ka con ex body such ha
•RKe−V(x)dx ≥1
2
•RK( (x)−Eµ (x))2dµ(x)≥1
2RRn( (x)−Eµ (x))2dµ(x),
•(EµK −Eµ )2≤Eµ|∇ |2
deno ing by µK he p obabili y suppo ed on Kwi h densi y
dµK(x) = e−V(x)dx
RKe−V(x)dx,
and aking in o accoun ha o any log-conca e p obabili y νone has ha he ope a o
no m o i s co a iance ma ix e i ies kCo νkop = sup
θ∈Sn−1Eνhx, θi2−(Eνhx, θi)2, we ob ain
Va µ ≤2√2Va µK + 2√2(EµK −Eµ )2≤2√2Va µK + 2√2Eµ|∇ |2
≤CnkCo µKkopEµK|∇ |2+ 2√2Eµ|∇ |2
=Cnsup
θ∈Sn−1EµKhx, θi2−(EµKhx, θi)2EµK|∇ |2+ 2√2Eµ|∇ |2
≤Cnsup
θ∈Sn−1
EµKhx, θi2EµK|∇ |2+ 2√2Eµ|∇ |2
≤4Cnsup
θ∈Sn−1
Eµhx, θi2Eµ|∇ |2+ 2√2Eµ|∇ |2
= (4Cn+ 2√2)Eµ|∇ |2≤5CnEµ|∇ |2.
90
The e o e, one can conside only compac ly suppo ed iso opic log-conca e p obabili ies
in Rnin o de o p o e (2).
By using a nice esul by E. Milman, [EM], in o de o p o e (2) i is enough o gi e an
uppe bound o he a iance o by an absolu e cons an imes k∇ k2
∞ o any Lipschi z
µin eg able unc ion. Besides, i is a 1-Lipschi z µ-in eg able unc ion one has
Va µ ≤Eµ| − (0)|2≤Eµ|x|2=n.
As a consequence one ob ains ha o e e y ixed n∈N, he alue o he cons an such
ha (2) holds o e e y log-conca e p obabili y µin Rnand Lipschi z µ-in eg able unc ion
is bounded by a cons an Cn, depending on N, The e o e, i is enough o p o e Theo em
1.3 o e e y n∈Nla ge han some ixed n0, since, changing he alue o he cons an C,
one can immedia ely ob ain he esul o e e y dimension n∈N.
In conclusion, one can conside only compac ly suppo ed log-conca e iso opic p oba-
bili ies in Rn o n≥n0 o some n0∈Nin o de o p o e (2).
We will include some p elimina y ac s o esul s we a e going o use.
Lemma 2.1. Le µbe any p obabili y on Rnand z∈Rn, hen
Eµhx−bµ, zi2=hAµz, zi.
P oo . Simply expand bo h exp essions.
P oposi ion 2.2 (Re e se Hölde ’s inequali y).The e exis s an absolu e cons an C > 0
such ha o e e y log-conca e p obabili y µon Rn, any semino m g:Rn→Rand 1≤
p≤qwe ha e
(Eµgp)1/p ≤(Eµgq)1/q ≤Cq
p(Eµgp)1/p .
P oo . See [BGVV, Theo em 2.4.6.],
The nex esul says ha we only need o ake in o accoun Bo el se s wi h p obabili y
1/2.
P oposi ion 2.3. Le µbe an iso opic log-conca e p obabili y on Rn. Assume ha he e
exis wo posi i e numbe s Θ, C > 0such ha
µ(EΘ E)≥C
91
o any Bo el se E∈Rnsuch ha µ(E) = 1
2, whe e EΘis he Θ-dila ion o E, i.e.
EΘ={e+x∈Rn:e∈E, |x|<Θ}.Then
µ+(A)≥C
Θmin{µ(A), µ(Ac)} o any Bo el se A⊂Rn.
P oo . See [EM2].
In o de o con ol he p obabili y o dila ions o Bo el se s, he ollowing concen a ion
esul s o mo e con ex han Gaussian p obabili ies can be applied
P oposi ion 2.4. Le φbe a con ex unc ion φ:Rn→Rand le > 0. Assume ha
dµ(x) = e−φ(x)−
2|x|2dx,
is a cen e ed p obabili y on Rn. Then o e e y Bo el se A⊂Rnsuch ha
1
10 ≤µ(A)≤9
10
we ha e
µAD
√ ≥95
100,
whe e D > 0is a sui ably chosen absolu e cons an independen o e e y o he pa ame e
and AD/√ is he D/√ -dila ion o A.
The p oo o his ac ollows om [BGVV, Theo em 14.6.6] (see also [AB1, Theo em
3.8]).
Nex we a e going o desc ibe some esul s on I ô p ocesses we a e going o use. (see
o ins ance, [O], [Kle]).
Le (Ω,F,P)a p obabili y space and (F ) ∈[0,T ]a il a ion in Ω, i.e., a amily o sub-σ-
algeb as on Ωsuch ha F 1⊆ F 2⊆ F, whene e 0≤ 1≤ 2≤T.
A one-dimensional I ô p ocess (X( )) ∈[0,T ]on Ωis a eal s ochas ic p ocess ha ing he
o m
X( ) = X(0) + Z
0
U(s)ds +Z
0
V(s)dW(s),0≤ ≤T,
whe e X(0) is F0-measu able and he p ocesses U( )and V( )a e F -adap ed and such ha
EPRT
0|U( )|d < ∞,EPRT
0V2( )d < ∞, and (W( )) ≥0is a Wiene p ocess (o B ownian
92
mo ion). I is said ha he p ocess (X( )) ∈[0,T ]has he s ochas ic di e en ial on [0, T]
dX( ) = U( )d +V( )dW( ),0≤ ≤T.
The p ocess (U( )) ∈[0,T ]is called he d i and (V( )) ∈[0,T]is called he di usion o
(X( )) ∈[0,T]. No e ha he p ocesses (U( )) ∈[0,T ]and (V( )) ∈[0,T ]may (and o en do)
depend on (X( )) ∈[0,T ]o he Wiene p ocess (W( )) ≥0as well.
In he case ha he p ocesses (U( )) ∈[0,T ]is Rn- alued, (V( )) ∈[0,T]is an (n×n)ma ix
and (W( )) ≥0is an n-dimensional Wiene p ocess, we say ha Xis an n-dimensional I ô
p ocess.
Le (X1( )) ∈[0,T],(X2( )) ∈[0,T ]be wo 1-dimensional I ô p ocesses. The quad a ic co-
a ia ion o [X1, X2] is de ined by
[X1, X2] = lim
kPk→0
N
X
k=0 X1(τk+1)−X1(τk)X2(τk+1)−X2(τk)
whe e P={0 = τ0≤τ1≤ ··· ≤ τN≤T}is a s ochas ic pa i ion o he non-nega i e
eal numbe s, kPk= max(τn−τn−1)is called he mesh o Pand he limi is de ined using
con e gence in p obabili y. I X2=X1we will deno e [X1] := [X1, X1] o e e y ∈[0, T].
In he case whe e dXi( ) = Ui( )d +hVi( ), dW( )i, o i= 1,2, whe e (Ui( )) ∈[0,T]and
(Vi( )) ∈[0,T](i= 1,2), a e n-dimensional adap ed s ochas ic p ocesses and (W( )) ≥0is an
n-dimensional Wiene p ocess [X1, X2] is also an I ô p ocess wi hou di usion and
(7) d[X1, X2] =hV1( ), V2( )id .
P oposi ion 2.5 (I ô’s o mula).Le (X( )) ∈[0,T ]be an n-dimensional I ô p ocess gi en
by dX( ) = U( )d +V( )dW( ), whe e U( )∈Rn,V( )is an n×nma ix and W( )is a
n-dimensional Wiene p ocess. Le g:Rn→Rbe a unc ion wi h g∈ C2)(Rn). Then he
s ochas ic p ocess (Y( )) ∈[0,T ]gi en by Y( ) = g(X( )) e i ies
dY ( ) = dg(X1( ), . . . , Xn( )) =
n
X
i=1
∂
∂xi
g(X1( ), . . . , Xn( ))dXi( )
+1
2
n
X
i,j=1
∂2
∂xi∂xj
g(X1( ), . . . , Xn( ))d[Xi, Xj] .
93
whe e δ is an adap ed, wi h bounded a ia ion p ocess, such ha
δ ≤(C p2σ2
nlog nT (Ap
)1+ 1
p,i p≥3
CT (A2
)3/2,i p= 2
and
| | ≤ CpT (Ap
)1+ 1
2p∀p≥2,
whe e C > 0is an absolu e cons an and σ2
n= sup E|X|−√n
2and he sup uns o e all
iso opic log-conca e andom ec o s in Rn.
P oo . We ollow Eldan’s me hod o compu e d(T (Ap
)). Howe e , o p= 2 we will use
he idea gi en by Lee-Vempala. In o de o do ha we will exp ess A in e ms o a special
o hono mal basis.
Le 0≥0be a ixed ime. Le ( i)n
i=1 be an o hono mal basis composed by he eigen-
ec o s o A 0and (αii( 0))n
i=1 he co esponding eigen alues. Assume ha he o hono mal
basis ( i)n
i=1 is o de ed in such way ha α11( 0)≥α22( 0)≥ ··· ≥ αnn( 0). Le also, o
any ≥0,αi,j =αi,j( ) := hA i, ji. We can exp ess, o any ≥0,
A =
n
X
i,j=1
αij i⊗ j.
I is no di icul o see ha o any na u al numbe p≥2and o any ≥0,
T (Ap
) = Xαi1i2αi2i3. . . αipi1,
whe e he sum uns o e all indices i1,...ip∈ {1, . . . , n}. No ice ha i = 0 hen
αij( 0) = hA 0 i, ji=δij, he K onecke del a. The e o e, di e en ia ing a = 0,
d(T (Ap
))| = 0=Xd(αi1i2αi2i3. . . αipi1) = 0
=
n
X
i=1
d(αp
ii)| = 0+X
i6=j
k1+k2+k3=p−2
d(αk1
ii αijαk2
jj αjiαk3
ii ) = 0
=
n
X
i=1
d(αp
ii)| = 0+X
i6=j
0≤k≤p−2
d(αk
iiαp−k−2
jj α2
ij) = 0
100

( he es o he e ms a e 0by I ô’s o mula). Acco ding o he exp ession o d(A )we ha e
ha
d(αij)| = 0=hd(A )| = 0 i, ji
=hEµ 0hx−b 0, iihx−b 0, ji(x−b 0), dW i−hA2
0 i, jid
=hξij, dW i−hA 0 i, A 0 jid =hξij, dW i−αiiαjjδijd ,
whe e ξij a e he ec o s ξij =ξi,j( 0) = Eµ 0hx−b 0, iihx−b 0, ji(x−b 0)∈Rn
By I ô’s o mula we ob ain he ollowing es ima es
d(αp
ii)| = 0=pαp−1
ii dαii| = 0+1
2p(p−1)αp−2
ii d[αii] 0
=1
2p(p−1)αp
ii |ξii|2
α2
ii −pαp+1
ii d +pαp
ii ξii
αii
, dW 
and o i<jand 0≤k≤p−2, since αij = 0 and αii ≥αjj,
d((αii)k(αjj)p−k−2(αij)2) = (αii)k(αjj)p−k−2d[αij]
= (αii)k+1(αjj)p−k−1|ξij|2
αiiαjj
d ≤(αii)p|ξij|2
αiiαjj
d .
The e o e, T (Ap
)is an I ô p ocess wi h
d(T (Ap
)) = δ d +h , dW i,
whe e o any = 0
δ 0=1
2p(p−1)
n
X
i=1
(αii)p|ξii|2
(αii)2−p
n
X
i=1
(αii)p+1
+X
i6=j
0≤k≤p−2
(αii)k+1(αjj)p−k−1|ξij|2
αiiαjj
and
0=p
n
X
i=1
αp
ii
ξii
αii
.
I is now enough o bound o m abo e δ and | |a each pa icula = 0. Fi s o all
101
we es ima e | |.
By using Cauchy-Schwa z and Bo ell’s e e se Hölde inequali ies (P oposi ion 2.2),
he e exis s an absolu e cons an C > 0such ha o e e y 1≤i≤n
|ξii|=ξii,ξii
|ξii|=Eµ hx−b , ii2x−b ,ξii
|ξii|
≤qEµ hx−b , ii4sEµ x−b ,ξii
|ξii|2
≤CEµ hx−b , ii2sEµ x−b ,ξii
|ξii|2
Taking in o accoun ha , by Lemma 2.1,
Eµ hx−b , zi2=hA z, zi ∀z∈Rn
we ob ain ha o e e y 1≤i≤n
|ξii| ≤ ChA i, iiA
ξii
|ξii|,ξii
|ξii|1/2
≤CαiikA k1/2
op .
Hence
| | ≤ CpkA k1/2
op T (Ap
)≤Cp(T (Ap
))1+1/(2p),
o some absolu e cons an C > 0.
Nex we will es ima e δ
i) Case p= 2
No e ha his necessa ily implies k= 0, and so we ha e a simple exp ession o δ ,
δ =
n
X
i,j=1 |ξij|2−2
n
X
i=1
(αii)3.
102
Thus, using again Bo ell’s e e se Hölde ’s inequali y wice
δ ≤
n
X
i,j=1 |ξij|2=
n
X
i,j=1 Eµ hx−b , iihx−b , ji(x−b )
2
=
n
X
i,j,k=1 |Eµ hx−b , iihx−b , jihx−b , ki|2
=Eµ ,x⊗µ ,y hx−b , y −b i3≤CEµ ,x Eµy, hx−b , y −b i23/2
=CEµ ,x hA (x−b ), x −b i3/2=CEµ ,x |A1/2
(x−b )|3
≤C2Eµ ,x |A1/2
(x−b )|23/2≤C2Eµ ,x hA (x−b ), x −b i3/2
=C2(T (A2
))3/2.
ii) Case p≥3. Now
δ ≤1
2p(p−1)
n
X
i=1
(αii)p|ξij|2
(αii)2+p(p−1) X
1≤i<j≤n
(αii)p|ξij|2
αiiαjj
≤p(p−1)
n
X
i=1
(αii)p
n
X
j=1
|ξij|2
αiiαjj
.
Le us ix 1≤i≤n. Then
n
X
j=1
|ξij|2
αiiαjj
=
n
X
j=1 
Eµ x−b , i
√αii x−b , j
√αjj (x−b )
2
= (?)
We pe o m in he in eg al de ining he expec a ion he change o a iables x−b =A1/2
y.
The in eg al wi h espec o he new a iable ycan be ega ded as an expec a ion wi h
espec o a p obabili y ν , which is iso opic and, since he ec o s (ηi)n
i=1 wi h ηi=
103
A1/2
i/√αii o m an o hono mal basis, we ha e
(?) =
n
X
j=1 
Eν A1/2
y, i
√αii A1/2
y, j
√αjj A1/2
y
2
=
n
X
j=1 A1/2
(Eν hy, ηiihy, ηjiy)
2
≤
n
X
j=1 kA1/2
k2
op |Eν hy, ηiihy, ηjiy|2
≤ kA kop sup
θ∈Sn−1
n
X
j=1 |Eν hy, θihy, ηjiy|2
=kA kop sup
θ∈Sn−1kEν y⊗yhy, θik2
HS
Thus
δ ≤C p2kA kopT (Ap
) sup
θ∈Sn−1kEν y⊗yhy, θik2
HS
Eldan p o ed in [E1, Lemma 1.6] ha he exp ession be o e is bounded om abo e by
sup
θ∈Sn−1kEν y⊗yhy, θik2
HS ≤Cσ2
nlog n
which gi es us he co esponding es ima e.
P oposi ion 5.2. The e exis n0∈Nand C > 0such ha o any n≥n0we ha e
P{kA kop ≤4,∀ ∈[0, T]}>0.9, o T=1
4C(log n)2σ2
n
and
P(kA kop ≤√51
7√n, ∀ ∈[0, T])≥0.9, o T=1
256C√n
P oo . Le p≥2, ixed. Conside he unc ion Φ( ) = −(n+T (Ap
))−1/p. Then Φ( )is an
104
I ô P ocess and
d(Φ( )) = 1
p
d(T (Ap
))
(n+T (Ap
))1+1/p −1
2p1 + 1
pd[T (Ap
)]
(n+T (Ap
))2+1/p
=1
p
δ
(n+T (Ap
))1+1/p −1 + p
2p2| |2
(n+T (Ap
))2+1/p d
+
p(n+T (Ap
))1+1/p , dW =α d +dZ ,
whe e α is an adap ed p ocess o bounded a ia ion and Z a ma ingale e m wi h Z0= 0.
By he p eceding P oposi ion δ ≤Lp(T (Ap
))1+1/p whe e Lpis a di e en exp ession
depending on whe he p= 2 o p≥3. The e o e,
α ≤Lp(T (Ap
))1+1/p
p(n+T (Ap
))1+1/p ≤Lp
p=(C p σ2
nlog n, i p≥3
C, i p= 2..
The quad a ic a ia ion o Z is
d[Z] =| |2
p2(n+T (Ap
))2+2/p d ≤C(T (Ap
))2+1/p
(n+T (Ap
))2+2/p d ≤C
n1/p d .
Then
Φ( )−Φ(0) = Z
0
αsds +Z ∀ ≥0.
We ix T > 0, hen
max
0≤ ≤TΦ( ) + (2n)−1/p ≤Lp
pT+ max
0≤ ≤TZ .
By he Dambis and Dubins-Schwa z heo em (see P oposi ion 2.7) we know ha Z is equal
in law o a B ownian mo ion ˜
W[Z] , so o any γ > 0we ha e, by he e lec ion p inciple
(see P oposi ion 2.6), ha
Pmax
0≤ ≤TΦ( ) + (2n)−1/p −Lp
pT > γ≤Pmax
0≤ ≤TZ > γ
=Pmax
0≤ ≤T
˜
W[Z] > γ≤Pmax
0≤s≤CT n−1/p
˜
Ws> γ
= 2Pn˜
WCT n−1/p > γo≤2 exp −γ2
2CTn−1/p 
105

We ake γ=1
4n1/p and T=p
256Lpn1/p and we achie e
Pmax
0≤ ≤TΦ( )> n−1/p 1
4+1
256 −2−1/p≤2 exp −8Lp
pC 
Since p≥2, we ha e 1
4<2−1/p −1
4−1
256 <7
10 and hen
Pmax
0≤ ≤TΦ( )>−7
10n−1/p≤2 exp −8Lp
pC .
We ema k ha
max
0≤ ≤TΦ( )>−7
10n−1/p ⇐⇒ max
[0,T]T (Ap
)≥10
7p
−1n
Hence we ob ain ha
Pmax
[0,T]T (Ap
)≥10
7p
−1n≤2 exp −8Lp
pC 
E en ually we will conside wo alues o pin o de o ge ou esul . On he one hand, i
we choose p= 2 we ha e T=1
256C√nand
Pmax
0≤ ≤TT (A2
)>51
49 n≤2 exp (−8)
and
P(max
0≤ ≤TkA kop >√51
7√n)≤2 exp (−8)
On he o he hand, i we choose p= log n. Then T=1
256Cσ2
n(log n)2and
P(max
0≤ ≤TT (Alog n
)> 10
7log n
−1!n)≤2 exp −8σ2
n(log n)2
106
No e ha , conside ing he Gaussian dis ibu ion, we ob ain σ2
n≥1/2and hen
P(max
0≤ ≤TT (Alog n
)>10
7log n
n)≤2 exp −2(log n)2
and
Pmax
0≤ ≤TkA kop >10e
7≤2 exp −2(log n)2
6 Gluing he es ima es
P oposi ion 6.1. The e exis s n0∈Nsuch ha i n≥n0,µis an iso opic log-conca e
p obabili y measu e on Rnand Eis a Bo el se µ(E) = 1
2, gi en he sys em o s ochas ic
di e en ial equa ions (8),µ be he measu e de ined by (9),g( ) = µ (E), and
T=1
4C σ2
n(log n)2o T =1
256C√n,
hen we ha e ha
P
g(T)−1
2
>1
4≤0.2.
P oo . We know ha
g(T)−1
2=g(T)−g(0) = ZT
0
dg( ) = ZT
0hη , dW i
whe e η =ZE
(x)(x−b )dx, being he densi y o he p obabili y measu e µ .
The unc ion g( )is a ma ingale and so, by Dambis, Dubins-Schwa z heo em, P opo-
si ion 2.7, we ha e ha in dis ibu ion
g(T)−g(0) = ¯
W[g]T, ≥0
whe e ¯
Wsis a Wiene p ocess and [g]Tis he quad a ic a ia ion o g, which is,
[g]T=ZT
0|η |2d .
107
Hence, o any M > 0,
P{|g(T)−1/2|>1/4}=P{| ¯
W[g]T|>1/4} ≤ P{[g]T> M}+Pmax
0≤ ≤M|¯
W |>1
4.
We will bound bo h summands om abo e. Taking in o accoun ha o e e y ≥0
|η |=η ,η
|η |=ZE
(x)(x−b ),η
|η |dx
≤sEµ (x−b ),η
|η |2
=sA
η
|η |,η
|η |≤qkA kop
we ha e ha
[g]T≤ZT
0kA kop d ≤Tmax
0≤ ≤TkA kop.
and hen
(12) P{[g]T> M} ≤ Pmax
0≤ ≤TkA kop >M
T
On he o he hand, −¯
W  ≥0is also a B ownian mo ion and hen we ha e
Pmax
0≤ ≤M|¯
W |>1
4≤Pmax
0≤ ≤M
¯
W >1
4+Pmax
0≤ ≤M−¯
W >1
4
= 4P¯
WM>1
4≤4 exp −1
32M.
(13)
We conside now wo cases:
In he case T=1
4Cσ2
n(log n)2we choose M= 4Tand hen, by (12) and P oposi ion
5.2
P{[g]T> M} ≤ Pmax
0≤ ≤TkA kop >1
4≤0.1.
and by (13)
Pmax
0≤ ≤M|¯
W |>1
4≤4 exp −C2σ2
n(log n)2
32 ≤0.1
o nla ge enough.
108
In he case T=1
256C√nwe choose M=1
128C. Then by (12) and P oposi ion 5.2
P{[g]T> M} ≤ Pmax
0≤ ≤TkA kop >2√n≤0.1
and by (13)
Pmax
0≤ ≤M|¯
W | ≤ 4≤exp (−4C)≤0.1,
assuming ha C > 2, which we can assume wi hou loss o gene ali y.
The la e esul , oge he wi h he discussion in Sec ion 4, gi e he p oo o Theo em
1.3.
Acknowledgmen s. This wo k is pa ially suppo ed by MINECO/MICINN p ojec s
MTM2016-77710-P, PID2019-105979GB-I00 and DGA p ojec E48_20R.
Re e ences
[AB1] D. Alonso-Gu ié ez, J. Bas e o App oaching he Kannan-Lo ász-Simono i s and
a iance conjec u es. Lec u e No es in Ma hema ics, Sp inge . Monog aph. 2131, (2015).
[AB2] D. Alonso-Gu ié ez, J. Bas e o Sob e la conje u a de sal o espec al de Kannan,
Lo ász y Simono i s. Gace a de la Real Sociedad Ma emá ica Española 23 no. 1 (2020),
49–64.
[BK] S. Bobko , A. Koldobsky,On he cen al limi p ope y o con ex bodies, Geome ic
aspec s o unc ional analysis. Lec u e No es in Ma h. 1807, 44–52. (Sp inge , Be lin, 2003)
[B] C. Bo ell,Con ex measu es on locally con ex spaces, A k. Ma h. 12 (1974), 239–252.
[BGVV] S. B azi ikos, A. Giannopoulos, P. Vale as, B. H. V i siou,Geome y o
Iso opic Con ex Bodies, Ma hema ical Su eys and Monog aphs 196 (Ame ican Ma hema -
ical Socie y, P o idence, RI., 2014)
[E1] R. Eldan,Thin shell implies spec al gap up o polylog ia a s ochas ic localiza ion scheme.
Geom. Func . Anal. (GAFA) 23 (2013), 532–569
109