Related papers: Remarks on the conjectured log-Brunn-Minkowski ine…

In ["Illumination of convex bodies with many symmetries", Mathematika 63 (2017)], Tikhomirov verified the Hadwiger-Boltyanski Illumination Conjecture for the class of 1-symmetric convex bodies of sufficiently large dimension. We propose an…

We show that for all $A, B \subseteq \{0,1,2\}^{d}$ we have $$ |A+B|\geq (|A||B|)^{\log(5)/(2\log(3))}. $$ We also show that for all finite $A,B \subset \mathbb{Z}^{d}$, and any $V \subseteq\{0,1\}^{d}$ the inequality $$ |A+B+V|\geq…

$L_p$ 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-\lambda)\cdot K +_p…

We prove the following isoperimetric type inequality: Given a finite absolutely continuous Borel measure on ${\mathbb R}^n$, halfspaces have maximal measure among all subsets with prescribed barycenter. As a consequence, we make progress…

In this article we study two classical potential-theoretic problems in convex geometry corresponding to a nonlinear capacity, $\mbox{Cap}_{\mathcal{A}}$, where $\mathcal{A}$-capacity is associated with a nonlinear elliptic PDE whose…

It is shown that each monotone Minkowski endomorphism of convex bodies gives rise to an isoperimetric inequality which directly implies the classical Urysohn inequality. Among this large family of new inequalities, the only affine invariant…

Let $\mu_p$ be the generalized Gaussian distribution on $\mathbb{R}^n$ with density $e^{-\frac{|x|^p}{p}}$ multiplied by a constant depending on $p\ge 1$ and $n$, and $\alpha_p(n)$ be the largest number such that the Brunn-Minkowski type…

Using an optimal containment approach, we quantify the asymmetry of convex bodies in $\mathbb{R}^n$ with respect to reflections across affine subspaces of a given dimension. We prove general inequalities relating these ''Minkowski…

We give an alternative look at the log-Sobolev inequality (LSI in short) for log-concave measures by semigroup tools. The similar idea yields a heat flow proof of LSI under some quadratic Lyapunov condition for symmetric diffusions on…

The Ehrhard-Borell inequality is a far-reaching refinement of the classical Brunn-Minkowski inequality that captures the sharp convexity and isoperimetric properties of Gaussian measures. Unlike in the classical Brunn-Minkowski theory, the…

This work is concerned with a P\'olya-Szeg\"o type inequality for anisotropic functionals of Sobolev functions. The relevant inequality entails a double-symmetrization involving both trial functions and functionals. A new approach that…

We address an old open question in convex geometry that dates back to the work of Minkowski: what are the equality cases of the monotonicity of mixed volumes? The problem is equivalent to that of providing a geometric characterization of…

In this paper we explore questions regarding the Minkowski sum of the boundaries of convex sets. Motivated by a question suggested to us by V.~Milman regarding the volume of $\partial K+ \partial T$ where $K$ and $T$ are convex bodies, we…

The support measures of a convex body are a common generalization of the curvature measures and the area measures. With respect to the Hausdorff metric on the space of convex bodies, they are weakly continuous. We provide a quantitative…

We present an argument which leads from the Brunn-Minkowski inequality to a Poincare' type inequality on the boundary of convex bodies with smooth boundary and positive Gauss curvature

We find a new sharp trace Gagliardo-Nirenberg-Sobolev inequality on convex cones, aswell as a weighted sharp trace Sobolev inequality on epigraphs of convex functions. This is done by using a generalized Borell-Brascamp-Lieb inequality,…

In this paper we revisit the anisotropic isoperimetric and the Brunn-Minkowski inequalities for convex sets. The best known constant $C(n)=Cn^{7}$ depending on the space dimension $n$ in both inequalities is due to Segal [\ref{bib:Seg.}].…

We consider a functional $\mathcal F$ on the space of convex bodies in $\R^n$ defined as follows: ${\mathcal F}(K)$ is the integral over the unit sphere of a fixed continuous functions $f$ with respect to the area measure of the convex body…

$L_p$-Christoffel-Minkowski problem arises naturally in the $L_p$-Brunn-Minkowski theory. It connects both curvature measures and area measures of convex bodies and is a fundamental problem in convex geometric analysis. Since the lack of…

We prove a conjecture by Ruzsa from 2006 on a discrete version of the Brunn-Minkowski inequality, stating that for any $A,B\subset\mathbb{Z}^k$ and $\epsilon>0$ with $B$ not contained in $n_{k,\epsilon}$ parallel hyperplanes we have…