Monotonicity and concavity of integral functionals involving area measures of convex bodies

“…Variational argument. Infinitesimal versions of Brunn-Minkowski type inequalities have been considered and extensively studied in Bakry, Ledoux [1], Bobkov [3], [4], Colesanti [12], [13], Hug, Saorin-Gomez [14], Kolesnikov, Milman [23], [26], [27], Livshyts, Marsiglietti [15], [16], and many others.…”

Section: Proof Of Lemma 24mentioning

confidence: 99%

On the Gardner-Zvavitch conjecture: symmetry in the inequalities of Brunn-Minkowski type

Kolesnikov¹,

Livshyts²

2018

Preprint

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In this paper, we study the conjecture of Gardner and Zvavitch from [21], which suggests that the standard Gaussian measure γ enjoys 1 n -concavity with respect to the Minkowski addition of symmetric convex sets. We prove this fact up to a factor of 2: that is, we show that for symmetric convex K and L, and λ ∈ [0, 1],More generally, this inequality holds for convex sets containing the origin. Further, we show that under suitable dimension-free uniform bounds on the Hessian of the potential, the log-concavity of even measures can be strengthened to p-concavity, with p > 0, with respect to the addition of symmetric convex sets.

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“…The Lemma of Cheng and Yau admits the following extension (see Lemma 2.3 in [5]). Note that we adopt the summation convention over repeated indices.…”

Section: Preliminariesmentioning

confidence: 99%