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The main goal of this paper is to present a series of inequalities connecting the surface area measure of a convex body and surface area measure of its projections and sections. We present a solution of a question from S. Campi, P.…

The first goal of this paper is to improve some of the results in \cite{BCPR}. Namely, we establish the $L_p$-Brunn-Minkwoski inequality for intrinsic volumes for origin-symmetric convex bodies that are close to the ball in the $C^2$ sense…

The Heckman-Opdam hypergeometric functions of type BC extend classical Jacobi functions in one variable and include the spherical functions of non-compact Grassmann manifolds over the real, complex or quaternionic numbers. There are various…

The classical Crofton formula explains how intrinsic volumes of a convex body $K$ in $n$-dimensional Euclidean space can be obtained from integrating a measurement function at sections of $K$ with invariantly moved affine flats. Motivated…

Given an affine variety X and a finite dimensional vector space of regular functions L on X, we associate a convex body to (X, L) such that its volume is responsible for the number of solutions of a generic system of functions from L. This…

The Kervaire Invariant 1 Problem until recently was an open problem in algebraic topology. Hill-Hopkins-Ravenel theorem clams a negative solution of the problem for all dimensions $n=2^l-2$, $l \ge 8$. We prove the statement of…

A generalization of Lozanovskii's result is proved. Let E be $k$-dimensional subspace of an $n$-dimensional Banach space with unconditional basis. Then there exist $x_1,..,x_k \subset E$ such that $B_E \p \subset \p absconv\{x_1,..,x_k\}$…

We prove several discrete Gagliardo-Nirenberg-Sobolev and Poincar\'e-Sobolev inequalities for some approximations with arbitrary boundary values on finite volume meshes. The keypoint of our approach is to use the continuous embedding of the…

In this paper we study approximations of functions of Sobolev spaces $W^2_{p,\loc}(\Omega)$, $\Omega\subset\mathbb R^n$, by Lipschitz continuous functions. We prove that if $f\in W^2_{p,\loc}(\Omega)$, $1\leq p<\infty$, then there exists a…

The notion of ball convexity, considered in finite dimensional real Banach spaces, is a natural and useful extension of usual convexity; one replaces intersections of half-spaces by suitable intersections of balls. A subset $S$ of a normed…

We consider abstract inverse problems between infinite-dimensional Banach spaces. These inverse problems are typically nonlinear and ill-posed, making the inversion with limited and noisy measurements a delicate process. In this work, we…

The lower dimensional Busemann-Petty problem asks whether origin symmetric convex bodies in $\mathbb{R}^n$ with smaller volume of all $k$-dimensional sections necessarily have smaller volume. As proved by Bourgain and Zhang, the answer to…

We study a variational problem whose critical point determines the Reeb vector field for a Sasaki-Einstein manifold. This extends our previous work on Sasakian geometry by lifting the condition that the manifolds are toric. We show that the…

We present here a new method for approximating functions defined on superreflexive Banach spaces by differentiable functions with $\alpha$-H\"older derivatives (for some $0<\alpha\leq 1$). The smooth approximation is given by means of an…

We show that given a real number $p<1$, a positive integer $n$ and a proper subspace $H$ of $\mathbb{R}^n$, the measure on the Euclidean sphere $\mathbb{S}^{n-1}$, which is concentrated in $H$ and whose restriction to the class of Borel…

The average section functional ${\rm as}(K)$ of a centered convex body in ${\mathbb R}^n$ is the average volume of central hyperplane sections of $K$: \begin{equation*}{\rm as}(K)=\int_{S^{n-1}}|K\cap \xi^{\perp }|\,d\sigma (\xi…

We give a positive solution for the hyperplane conjecture of quotient spaces F of $L_p$, where $1<p\kll\infty$. \[ vol(B_F)^{\frac{n-1}{n}} \kl c_0 \pl p' \pl \sup_{H \p hyperplane} vol(B_F\cap H) \pl.\] This result is extended to Banach…

We determine precisely when the Bergman projection $P_\beta$ is bound\-ed from Lebesgue spaces $L^p_\alpha$ to weighted Bergman spaces $\mathcal B^p_\alpha$ of $\mathcal H$-harmonic functions on the hyperbolic ball, and verify a recent…

We introduce in this document a direct method allowing to solve numerically inverse type problems for linear hyperbolic equations. We first consider the reconstruction of the full solution of the wave equation posed in $\Omega\times (0,T)$…

We study convex sets C of finite (but non-zero volume in Hn and En. We show that the intersection of any such set with the ideal boundary of Hn has Minkowski (and thus Hausdorff) dimension of at most (n-1)/2, and this bound is sharp. In the…