On the Gaussian Brunn-Minkowski inequalityGalyna Livshyts, Fedor Nazarov

Kent State University

Preliminaries

The classical Brunn-Minkowski inequalityThe classical Brunn-Minkowski inequality states that for any measurable sets A,B ⊂ Rn and for anyλ ∈ [0, 1],

|λA + (1− λ)B|1n ≥ λ|A|

1n + (1− λ)|B|

1n, (1)

where | · | stands for the Lebesgue Measure on Rn. Here

K +Q = {a + b | a ∈ K, b ∈ Q}

is the Minkowski sum of the sets K and Q

andλA := {λa | a ∈ A}.

The standard Gaussian measure

The standard Gaussian Measure γ2 on Rn is the measure with density(

1√2π

)ne−|y|22 .

The support function of a convex setA set Q in Rn is called convex if together with every two points x, y ∈ Q it contains the whole segment[x, y].

The support function hQ of a convex set Q ⊂ Rn is the function on the unit sphere defined byhQ(θ) = maxx∈Q〈x, θ〉. By homogeneity it extends from the sphere to the whole space. The supportfunction represents the distance from the origin to the support hyperplane of a convex set in a given direc-tion:

The gradient of the support function of Q at θ is the vector with the endpoint in the contact point of thesupport hyperplane and Q.

The Gaussian Brunn-Minkowski inequalityGardner and Zvavitch conjectured in [4] that for the standard Gaussian measure γ2 the inequality analogousto (1) holds under some natural assumptions on the sets A and B in Rn:

γ2(λA + (1− λ)B)1n ≥ λγ2(A)

1n + (1− λ)γ2(B)

1n. (2)

Some positive results about the inequality (2) were obtained in [4] and later generalized by Marsiglietti [5].However, (2) is clearly false in the full generality: one may shift the set A away from the origin. The fartherthe shift, the smaller the right hand side of (2), while the left hand side is bounded from below by the fixedquantity (1− λ)γ2(B)

1n.

That gives a clue, which assumptions must be reinforced in order for (2) to hold.

The inequality (2) holds true if:• The dimension n = 1 (Gardner, Zvavitch [4]);• The Gaussian measure γ2(λA+ (1− λ)B) is larger then a certain constant (observed by Klartag; follows

from Erchard’s inequality);• γ2(A) = γ2(B) (is entailed by log-concavity of γ2);• The sets A and B are dilates of each other, i.e. A = x ·B for a positive number x (Gardner, Zvavitch [4],

follows from B-Theorem [3]).

Question 1 from [4]Does the inequality (2) hold true for all convex sets A and B containing the origin?

The answer is NO (obtained by Tkozh, Nayar [6]). Their counterexample looks roughly like this:

Question 2 from [4], [6]Does the inequality (2) hold true for all convex origin-symmetric sets A and B?

This Question is still OPEN!

The recent advances

The ideaPick a positive number a. Let h(u) be a strictly convex C2−smooth function on the sphere Sn−1. Considera function ψ(u) ∈ C2(Sn−1). Let s ∈ [0, a]. Consider

Kn(h(u), ψ(u), a) := {Ks}as=0 (3)

– a family of sets Ks in Rn, where the support function of each Ks is hs = h + sψ.

The approach used to attack this problem was inspired by the work of Colesanti, Hug, Saorin-Gomez [1],[2]: the inequality (2) is equivalent to concavity of the function f (s) := n

√γ2(Ks) in s.

Formula expressing the Gaussian measure in terms of the support functionIn order to compute anything using this idea, one needs a formula expressing a measure of a set in terms ofits support function. The formula is stated in the nextProposition 0.1. Let γ be a measure in Rn with density f (x). LetK be a strictly convex body in Rn contain-ing the origin with the support function h(u) ∈ C2(Sn−1). By detQ(h(u)) denote the curvature function ofK. Denote the gradient of h by ∇h. Then

γ(K) =

∫Sn−1

h(u) detQ(h(u))

|∇h(u)|n

∫ |∇h|0

tn−1f(t · ∇h|∇h|

)dtdu.

For the standard Gaussian measure on the plane the above is simpler:Corollary 0.2. Let γ2 be the Standard Gaussian measure in R2. Let K be a strictly convex body in R2

containing the origin with the support function h(u) ∈ C2(S1). Then

γ2(K) = 1− 1

2π

∫ π

−π

h2 + hh

h2 + h2e−

h2+h2

2 du.

Corollary 0.2 can be rewritten in even simpler way: under the assumptions of Corollary 0.2,

γ2(K) =1

4π

∫ π

−π(h2 + hh)

∫ 1

0e−t

h2+h2

2 dtdu.

The inequality (2) holds in a neighborhood of any discThe first outcome from the method described above is the following

Proposition 0.3. Pick R ∈ (0,∞). Pick ψ ∈ C2(S1). Then there exists an ε = ε(R,ψ) such that for everyK,L ∈ K2(R,ψ, ε) (see (3) for the notation), and for every λ ∈ [0, 1],

γ122(λK + (1− λ)L) ≥ λγ

122(K) + (1− λ)γ

122(L).

Essentially, that means that when both of the setsA andB are close to a ball of any radiusR, the inequality(2) holds true.

The criterion for the Gaussian Brunn-Minkowski inequalityAlso, the method described above entails a criterion of the inequality (2) in terms of the support function.In particular, for the symmetric case the criterion is the following:

Proposition 0.4. The inequality (2) holds for every pair of convex symmetric sets A and B in R2 if and onlyif for any even functions h(u) ∈ C2[−1, 1] and ψ(u) ∈ C2[−1, 1] such that h(−1) = h(1), h(−1) = h(1),h(−1) = h(1), ψ(−1) = ψ(1), ψ(−1) = ψ(1) and ψ(−1) = ψ(1), and also h(u) + h(u) > 0 for allu ∈ [−1, 1], the following inequality holds:

2 ·

[∫ 1

−1

(ψ2(1− h(h + h))− ψ2

)e−

h2+h2

2 du

]·[∫ 1

−1

h2 + hh

h2 + h2(1− e−

h2+h2

2 )du

]≤ (4)(∫ 1

−1ψ(h + h)e−

h2+h2

2

)2

.

For polytopes the inequality (4) involves a finite set of parameters. The check is being run, whether theinequality (4) holds for symmetric polytopes; the collected data suggests that (4), and, consequently, (2)hold true for all symmetric sets.

References[1] A. Colesanti, From the Brunn-Minkowski inequality to a class of Poincare’ type inequalities, Communications in Contempo-

rary Mathematics, 10 n. 5 (2008), 765-772.

[2] A. Colesanti, D. Hug, E. Saorin-Gomez, A characterization of some mixed volumes vie the Brunn-Minkowski inequality,Journal of Geometric Analysis (2012), pp. 1-28.

[3] D. Cordero-Erausquin, A. M. Fradelizi, B. Maurey, The (B) conjecture for the Gaussian measure of dilates of symmetricconvex sets and related problems, Journal of Functional Analysis Vol. 214 (2004) 410-427.

[4] R. Gardner, A. Zvavitch Gaussian Brunn-Minkowski-type inequlities, Trans. Amer. Math. Soc., 360, (2010), 10, 5333-5353.

[5] A. Marsiglietti, On improvement of the concavity of convex measures, http://perso-math.univ-mlv.fr/users/marsiglietti.arnaud/index.html.

[6] P. Nayar, T. Tkocz, A Note on a Brunn-Minkowski Inequality for the Gaussian Measure, Proc. Amer. Math. Soc. 141 (2013),11, 4027-4030.