On a conjectural symmetric version of Ehrhard’s inequality
Authors: Livshyts Galyna 
Published: April 2024

We formulate a plausible conjecture for the optimal Ehrhard-type inequality for convex symmetric sets with respect to the Gaussian measure. Namely, letting

J

k
−
1

(
s
)
=

∫
0
s

t

k
−
1

e

−

t
2

2

d
t

J_{k-1}(s)=\int ^s_0 t^{k-1} e^{-\frac {t^2}{2}}dt

and

c

k
−
1

=

J

k
−
1

(
+
∞
)

c_{k-1}=J_{k-1}(+\infty )

, we conjecture that the function

F
:
[
0
,
1
]
→

R

F:[0,1]\rightarrow \mathbb {R}

, given by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F left-parenthesis a right-parenthesis equals sigma-summation Underscript k equals 1 Overscript n Endscripts 1 Subscript a element-of upper E Sub Subscript k Baseline dot left-parenthesis beta Subscript k Baseline upper J Subscript k minus 1 Superscript negative 1 Baseline left-parenthesis c Subscript k minus 1 Baseline a right-parenthesis plus alpha Subscript k Baseline right-parenthesis">

F
(
a
)
=

∑

k
=
1

n

1

a
∈

E
k

⋅
(

β
k

J

k
−
1

−
1

(

c

k
−
1

a
)
+

α
k

)

\begin{equation*} F(a)= \sum _{k=1}^n 1_{a\in E_k}\cdot (\beta _k J_{k-1}^{-1}(c_{k-1} a)+\alpha _k) \end{equation*}

(with an appropriate choice of a decomposition

[
0
,
1
]
=

∪

i

E
i

[0,1]=\cup _{i} E_i

and coefficients

α
i

,

β
i

\alpha _i, \beta _i

) satisfies, for all symmetric convex sets

K
K

and

L
L

, and any

λ
∈
[
0
,
1
]

\lambda \in [0,1]

,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F left-parenthesis gamma left-parenthesis lamda upper K plus left-parenthesis 1 minus lamda right-parenthesis upper L right-parenthesis right-parenthesis greater-than-or-equal-to lamda upper F left-parenthesis gamma left-parenthesis upper K right-parenthesis right-parenthesis plus left-parenthesis 1 minus lamda right-parenthesis upper F left-parenthesis gamma left-parenthesis upper L right-parenthesis right-parenthesis period">

F

(
γ
(
λ
K
+
(
1
−
λ
)
L
)
)

≥
λ
F

(
γ
(
K
)
)

+
(
1
−
λ
)
F

(
γ
(
L
)
)

.

\begin{equation*} F\left (\gamma (\lambda K+(1-\lambda )L)\right )\geq \lambda F\left (\gamma (K)\right )+(1-\lambda ) F\left (\gamma (L)\right ). \end{equation*}

We explain that this conjecture is “the most optimistic possible”, and is equivalent to the fact that for any symmetric convex set

K
K

, its Gaussian concavity power

p
s

(
K
,
γ
)

p_s(K,\gamma )

is greater than or equal to

p
s

(
R

B
2
k

×

R

n
−
k

,
γ
)

p_s(RB^k_2\times \mathbb {R}^{n-k},\gamma )

, for some

k
∈
{
1
,
…
,
n
}

k\in \{1,\dots ,n\}

. We call the sets

R

B
2
k

×

R

n
−
k

RB^k_2\times \mathbb {R}^{n-k}

round

k
k

-cylinders; they also appear as the conjectured Gaussian isoperimetric minimizers for symmetric sets, see Heilman [Amer. J. Math. 143 (2021), pp. 53–94].

In this manuscript, we make progress towards this question, and show that for any symmetric convex set

K
K

in

R

n

\mathbb {R}^n

,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p Subscript s Baseline left-parenthesis upper K comma gamma right-parenthesis greater-than-or-equal-to sup Underscript upper F element-of upper L squared left-parenthesis upper K comma gamma right-parenthesis intersection upper L i p left-parenthesis upper K right-parenthesis colon integral upper F equals 1 Endscripts left-parenthesis 2 upper T Subscript gamma Superscript upper F Baseline left-parenthesis upper K right-parenthesis minus upper V a r left-parenthesis upper F right-parenthesis right-parenthesis plus StartFraction 1 Over n minus double-struck upper E upper X squared EndFraction comma">

p
s

(
K
,
γ
)
≥

sup

F
∈

L
2

(
K
,
γ
)
∩
L
i
p
(
K
)
:

∫
F
=
1

(
2

T

γ

F

(
K
)
−
V
a
r
(
F
)
)

+

1

n
−

E

X
2

,

\begin{equation*} p_s(K,\gamma )\geq \sup _{F\in L^2(K,\gamma )\cap Lip(K):\,\int F=1} \left (2T_{\gamma }^F(K)-Var(F)\right )+\frac {1}{n-\mathbb {E}X^2}, \end{equation*}

where

T

γ

F

(
K
)

T_{\gamma }^F(K)

is the

F
−

F-

torsional rigidity of

K
K

with respect to the Gaussian measure. Moreover, the equality holds if and only if

K
=
R

B
2
k

×

R

n
−
k

K=RB^k_2\times \mathbb {R}^{n-k}

for some

R
>
0

R>0

and

k
=
1
,
…
,
n

k=1,\dots ,n

. As a consequence, we get
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p Subscript s Baseline left-parenthesis upper K comma gamma right-parenthesis greater-than-or-equal-to upper Q left-parenthesis double-struck upper E StartAbsoluteValue upper X EndAbsoluteValue squared comma double-struck upper E double-vertical-bar upper X double-vertical-bar Subscript upper K Superscript 4 Baseline comma double-struck upper E double-vertical-bar upper X double-vertical-bar Subscript upper K Superscript 2 Baseline comma r left-parenthesis upper K right-parenthesis right-parenthesis comma">

p
s

(
K
,
γ
)
≥
Q
(

E

|

X

|

2

,

E

‖
X

‖
K
4

,

E

‖
X

‖
K
2

,
r
(
K
)
)
,

\begin{equation*} p_s(K,\gamma )\geq Q(\mathbb {E}|X|^2, \mathbb {E}\|X\|_K^4, \mathbb {E}\|X\|^2_K, r(K)), \end{equation*}

where

Q
Q

is a certain rational function of degree

2
2

, the expectation is taken with respect to the restriction of the Gaussian measure onto

K
K

,

‖
⋅

‖
K

\|\cdot \|_K

is the Minkowski functional of

K
K

, and

r
(
K
)

r(K)

is the in-radius of

K
K

. The result follows via a combination of some novel estimates, the

L
2

L2

method (previously studied by several authors, notably Kolesnikov and Milman [J. Geom. Anal. 27 (2017), pp. 1680–1702; Amer. J. Math. 140 (2018), pp. 1147–1185; Geometric aspects of functional analysis, Springer, Cham, 2017; Mem. Amer. Math. Soc. 277 (2022), v+78 pp.], Kolesnikov and the author [Adv. Math. 384 (2021), 23 pp.], Hosle, Kolesnikov, and the author [J. Geom. Anal. 31 (2021), pp. 5799–5836], Colesanti [Commun. Contemp. Math. 10 (2008), pp. 765–772], Colesanti, the author, and Marsiglietti [J. Funct. Anal. 273 (2017), pp. 1120–1139], Eskenazis and Moschidis [J. Funct. Anal. 280 (2021), 19 pp.]), and the analysis of the Gaussian torsional rigidity.

As an auxiliary result on the way to the equality case characterization, we characterize the equality cases in the “convex set version” of the Brascamp-Lieb inequality, and moreover, obtain a quantitative stability version in the case of the standard Gaussian measure; this may be of independent interest. All the equality case characterizations rely on the careful analysis of the smooth case, the stability versions via trace theory, and local approximation arguments.

In addition, we provide a non-sharp estimate for a function

F
F

whose composition with

γ
(
K
)

\gamma (K)

is concave in the Minkowski sense for all symmetric convex sets.