Related papers: Stability inequalities for projections of convex b…

We review recent stability and separation results in volume comparison problems and use them to prove several hyper- plane inequalities for intersection and projection bodies.

We prove a hyperplane inequality for the surface area of projection bodies.

We provide general inequalities that compare the surface area S(K) of a convex body K in ${\mathbb R}^n$ to the minimal, average or maximal surface area of its hyperplane or lower dimensional projections. We discuss the same questions for…

We prove an estimate for arbitrary measure of sections of convex bodies. The proof is based on a stability result for intersection bodies.

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.…

We generalize the hyperplane inequality in dimensions up to 4 to the setting of arbitrary measures in place of the volume. To prove this generalization we establish stability in the affirmative part of the solution to the Busemann-Petty…

We prove stability in the affirmative part of the Busemann-Petty problem on sections of complex convex bodies.

We consider a generalization of the hyperplane problem to arbitrary measures in place of volume and to sections of lower dimensions. We prove this generalization for unconditional convex bodies and for duals of bodies with bounded volume…

We prove an inequality that extends to arbitrary measures the hyperplane inequality for volume of unconditional convex bodies originally observed by Bourgain.

A $\sqrt{n}$ estimate in the hyperplane problem with arbitrary measures has recently been proved in \cite{K3}. In this note we present analogs of this result for sections of lower dimensions and in the complex case. We deduce these…

Some inequalities for different types of convexity are established.

We prove that the static convexity is preserved along two kinds of locally constrained curvature flows in hyperbolic space. Using the static convexity of the flow hypersurfaces, we prove new family of geometric inequalities for such…

We prove several inequalities estimating the distance between volumes of two bodies in terms of the maximal or minimal difference between areas of sections or projections of these bodies. We also provide extensions in which volume is…

We prove a stability version of the isodiametric inequality on the sphere and in the hyperbolic space.

In this paper, we present sharp stability results for various reverse isoperimetric problems in $\mathbb R^2$. Specifically, we prove the stability of the reverse isoperimetric inequality for $\lambda$-convex bodies -- convex bodies with…

This paper contains a new concept to measure the width and thickness of a convex body in the hyperbolic plane. We compare the known concepts with the new one and prove some results on bodies of constant width, constant diameter and given…

In convex geometry, the constructions that assign to a convex body its difference body, projection body, or volume have the following properties: They are (1) invariant under volume-preserving linear changes of coordinates; (2) continuous;…

We show that the cone-volume measure of a convex body with centroid at the origin satisfies the subspace concentration condition. This implies, among others, a conjectured best possible inequality for the $\mathrm{U}$-functional of a convex…

This article is a survey of recent results on slicing inequalities for convex bodies. The focus is on the setting of arbitrary measures in place of volume.

In this paper we study two different weighted isoperimetric inequalities. In the first part of the paper we prove a sharp stability result for the isoperimetric inequality with a log-convex weight. In the second part we analize the behavior…