We investigate the $L_p$ Brunn-Minkowski inequality for dual quermassintegrals in weighted measure spaces, which is a special class of rotationally invariant measures proposed by Cordero-Erausquin and Rotem [Ann. Probab., {\bf 51} (2023)]. Specifically, the weighted dual quermassintegral is defined by integrating the radial density $|x|^{q-n}ϕ(|x|)$ for $q\in(0,n]$, where $ϕ$ is a positive radially non-increasing weight, it recovers the classical dual quermassintegral when $ϕ\equiv1$. For $p\geq1$, we prove the $L_p$ Brunn-Minkowski inequality with concavity exponent $p/q$ under the condition that $t\mapsto\logϕ(e^t)$ is concave, which is exactly the natural convexity condition from Cordero-Erausquin and Rotem's paper in general, improving the exponent $1/n$ when $p=1$. For $p\in(0,1)$, we obtain the result with exponent $p/q$ under more strictly weight assumptions, together with explicit lower bounds for the admissible range of $p$.
In this paper, we introduce first the mixed affine quermassintegrals. The Aleksandrov-Fenchel inequality for the mixed affine quermassintegrals is established. As an application, the Minkowski, Brunn-Minkowski inequalities for the mixed affine quermassintegrals are also derived.
We prove the validity of the $p$-Brunn-Minkowski inequality for the intrinsic volume $V_k$, $k=2,\dots, n-1$, of convex bodies in $\mathbb{R}^n$, in a neighborhood of the unit ball, for $0\le p<1$. We also prove that this inequality does not hold true on the entire class of convex bodies of $\mathbb{R}^n$, when $p$ is sufficiently close to $0$.
$L_p$ Brunn-Minkowski type inequa\-li\-ties for the lattice point enumerator $\mathrm{G}_n(\cdot)$ are shown, both in a geometrical and in a functional setting. In particular, we prove that \[\mathrm{G}_n\bigl((1-λ)\cdot K +_p λ\cdot L + (-1,1)^n\bigr)^{p/n}\geq (1-λ)\mathrm{G}_n(K)^{p/n}+λ\mathrm{G}_n(L)^{p/n}\] for any $K, L\subset\mathbb{R}^n$ bounded sets with integer points and all $λ\in(0,1)$. We also show that these new discrete analogues (for $\mathrm{G}_n(\cdot)$) imply the corresponding results concerning the Lebesgue measure.
We consider the problem of finding the best function $φ_n:[0,1]\to\mathbb{R}$ such that for any pair of convex bodies $K,L\in\mathbb{R}^n$ the following Brunn-Minkowski type inequality holds $$ |K+_θL|^\frac{1}{n}\geqφ_n(θ)(|K|^\frac{1}{n}+|L|^\frac{1}{n}), $$ where $K+_θL$ is the $θ$-convolution body of $K$ and $L$. We prove a sharp inclusion of the family of Ball's bodies of an $α$-concave function in its super-level sets in order to provide the best possible function in the range $\left(\frac{3}{4}\right)^n\leqθ\leq1$, characterizing the equality cases.
A classification of SL$(n)$ contravariant, continuous function valued valuations on convex bodies is established. Such valuations are natural extensions of SL$(n)$ contravariant $L_p$ Minkowski valuations, the classification of which characterized $L_p$ projection bodies, which are fundamental in the $L_p$ Brunn-Minkowski theory, for $p \geq 1$. Hence our result will help to better understand extensions of the $L_p$ Brunn-Minkowski theory. In fact, our results characterize general projection functions which extend $L_p$ projection functions ($p$-th powers of the support functions of $L_p$ proj…
In this paper, we prove a Prékopa-Leindler type inequality of the $L_p$ Brunn-Minkowski inequality. It extends an inequality proved by Das Gupta [8] and Klartag [16], and thus recovers the Prékopa-Leindler inequality. In addition, we prove a functional $L_p$ Minkowski inequality.
We present an alternative, short proof of a recent discrete version of the Brunn-Minkowski inequality due to Lehec and the second named author. Our proof also yields the four functions theorem of Ahlswede and Daykin as well as some new variants.
We give an alternative proof for discrete Brunn-Minkowski type inequalities, recently obtained by Halikias, Klartag and the author. This proof also implies somewhat stronger weighted versions of these inequalities. Our approach generalizes ideas of Gozlan, Roberto, Samson and Tetali from the theory of measure transportation and provides new displacement convexity of entropy type inequalities for the lattice point enumerator.
We study several of the recent conjectures in regards to the role of symmetry in the inequalities of Brunn-Minkowski type, such as the $L_p$-Brunn-Minkowski conjecture of Böröczky, Lutwak, Yang and Zhang, and the Dimensional Brunn-Minkowski conjecture of Gardner and Zvavitch, in a unified framework. We obtain several new results for these conjectures. We show that when $K\subset L,$ the multiplicative form of the $L_p$-Brunn-Minkowski conjecture holds for Lebesgue measure for $p\geq 1-Cn^{-0.75}$, which improves upon the estimate of Kolesnikov and Milman in the partial case when one body is co…
The aim of this note is to show that the local form of the logarithmic Brunn-Minkowski conjecture holds for zonoids. The proof uses a variant of the Bochner method due to Shenfeld and the author.
We extend Prekopa's Theorem and the Brunn-Minkowski Theorem from convexity to $F$-subharmonicity. We apply this to the interpolation problem of convex functions and convex sets introducing a new notion of "harmonic interpolation" that we view as a generalization of Minkowski-addition.
The curvature dimension condition CD(K,N), pioneered by Sturm and Lott--Villani, is a synthetic notion of having curvature bounded below and dimension bounded above, in the non-smooth setting. This condition implies a suitable generalization of the Brunn--Minkowski inequality, denoted by BM(K,N). In this paper, we address the converse implication in the setting of weighted Riemannian manifolds, proving that BM(K,N) is in fact equivalent to CD(K,N). Our result allows to characterize the curvature dimension condition without using neither the optimal transport nor the differential structure of t…
