Related papers: Log-Concavity Properties of Minkowski Valuations
We begin a systematic study of the region of possible values of the volumes of Minkowski subset sums of a collection of $M$ compact sets in $\mathbb{R}^d$, which we call the Lyusternik region, and make some first steps towards describing…
It is easy to show that the lower and the upper box dimensions of a bounded set in Euclidean space are invariant with respect to the ambient space. In this article we show that the Minkowski content of a Minkowski measurable set is also…
In this paper, we derive new sharp weighted Alexandrov-Fenchel and Minkowski inequalities for smooth, closed hypersurfaces under various convexity assumptions in Euclidean, spherical, and hyperbolic spaces. These inequalities extend…
This paper introduces the dual Orlicz-Brunn-Minkowski theory for star sets. A radial Orlicz addition of two or more star sets is proposed and a corresponding dual Orlicz-Brunn-Minkowski inequality is established. Based on a radial Orlicz…
In this short note we explain why the log-Brunn-Minkowski conjecture is correct for complex convex bodies. We do this by relating the conjecture to the notion of complex interpolation, and appealing to a general theorem by…
In this short survey we want to present some of the impact of Minkowski's successive minima within Convex and Discrete Geometry. Originally related to the volume of an $o$-symmetric convex body, we point out relations of the successive…
Elaborating on the similarity between the entropy power inequality and the Brunn-Minkowski inequality, Costa and Cover conjectured in {\it On the similarity of the entropy power inequality and the Brunn-Minkowski inequality} (IEEE Trans.…
Lidskii's additive inequalities (both for eigenvalues and singular values) can be interpreted as an explicit description of global minimizers of functions that are built on unitarily invariant norms, with domains consisting of certain…
We prove stability estimates for the Brunn-Minkowski inequality for convex sets. Unlike existing stability results, our estimates improve as the dimension grows. Our results are equivalent to a thin shell bound, which is one of the central…
The famous Hadwiger theorem classifies all rigid motion invariant continuous valuations on convex sets as linear conbinations of quermassintegrals. We prove much more general result. We classify continuous valuations which are invariant…
In this paper, we introduce operator geodesically convex and operator convex-log functions and characterize some properties of them. Then apply these classes of functions to present several operator Azc\'{e}l and Minkowski type inequalities…
We present a complete classification of $\operatorname{SL}(n)$ contravariant, $C(\mathbb{R}^n\setminus\{o\})$-valued valuations on polytopes, without any additional assumptions.It extends the previous results of the second author [Int.…
The main goal of this exposition is to present further analysis of the Kantorovich and Ando operator inequalities. In particular, a new proof of Ando's inequality is given, a new non-trivial refinement of Kantorovich inequality is shown,…
In this paper, combining the $p$-capacity for $p\in (1, n)$ with the Orlicz addition of convex domains, we develop the $p$-capacitary Orlicz-Brunn-Minkowski theory. In particular, the Orlicz $L_{\phi}$ mixed $p$-capacity of two convex…
The aim of this paper is to develop a basic framework of the $L_p$ theory for the geometry of log-concave functions, which can be viewed as a functional "lifting" of the $L_p$ Brunn-Minkowski theory for convex bodies. To fulfill this goal,…
We give a new approach, inspired by H\"ormander's $L^2$-method, to weighted variance inequalities which extend results obtained by Bobkov and Ledoux. It provides in particular a local proof of the dimensional functional forms of the…
We prove that the first (nontrivial) Dirichlet eigenvalue of the Ornstein-Uhlenbeck operator $$ L(u)=\Delta u-\langle\nabla u,x\rangle\,, $$ as a function of the domain, is convex with respect to the Minkowski addition, and we characterize…
The algebra of smooth translation-invariant valuations on convex bodies, introduced by S.Alesker in the early 2000s, was in part proved and in part conjectured to satisfy properties formally analogous to those of the cohomology ring of a…
We investigate quantitative implications of the notion of log-concavity through a probabilistic interpretation. In particular, we derive concentration inequalities, moment and entropy bounds for random variables satisfying a precise degree…
This paper establishes two new geometric inequalities in the dual Brunn-Minkowski theory. The first, originally conjectured by Lutwak, is the Brunn-Minkowski inequality for dual quermassintegrals of origin-symmetric convex bodies. The…