Related papers: Subspace concentration of dual curvature measures

We study subspace concentration of dual curvature measures of convex bodies $K$ satisfying $\gamma (-K)\subseteq K$ for some $\gamma \in (0,1]$. We present upper bounds on the subspace concentration depending on $\gamma$, which, in…

We prove tight subspace concentration inequalities for the dual curvature measures $\widetilde{\mathrm{C}}_q(K,\cdot)$ of an $n$-dimensional origin-symmetric convex body for $q\geq n+1$. This supplements former results obtained in the range…

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 paper studies the general Lp dual curvature density equation under a group symmetry assumption. This geometric partial differential equation arises from the general Lp dual Minkowski problem of prescribing the Lp dual curvature measure…

An affine version of the linear subspace concentration inequality as proposed by Wu is established for centered convex bodies. This generalizes results from Wu and Freyer, Henk, Kipp on polytopes to convex bodies.

We characterize the symmetric measures which satisfy the one dimensional convex infimum convolution inequality of Maurey. For these measures the tensorization argument yields the two level Talagrand's concentration inequalities for their…

We prove that for $s<0$, $s$-concave measures on ${\mathbb R}^n$ satisfy a thin shell concentration similar to the log-concave one. It leads to a Berry-Esseen type estimate for their one dimensional marginal distributions. We also establish…

The dual Minkowski problem for even data asks what are the necessary and sufficient conditions on an even prescribed measure on the unit sphere for it to be the $q$-th dual curvature measure of an origin-symmetric convex body in…

In this paper, we use the inverse curvature flow to prove a sharp geometric inequality on star-shaped and two-convex hypersurface in hyperbolic space.

We give a sharp lower bound on the area of the domain enclosed by an embedded curve lying on a two-dimensional sphere, provided that geodesic curvature of this curve is bounded from below. Furthermore, we prove some dual inequalities for…

We prove that an approximated version of the Brunn--Minkowski inequality with volume distortion coefficient implies a Gaussian concentration-of-measure phenomenon. Our main theorem is applicable to discrete spaces.

By means of dual convex bodies, we obtain regularity of solutions to the expanding Gauss curvature flows with homogeneity degrees $-p$, $0<p<1$. At the end, we remark that our method can also be used to obtain regularity of solutions to the…

The intrinsic volumes are measures of the content of a convex body. This paper uses probabilistic and information-theoretic methods to study the sequence of intrinsic volumes of a convex body. The main result states that the intrinsic…

We introduce dual curvature measures for log-concave functions, which in the case of characteristic functions recover the dual curvature measures for convex bodies introduced by Huang-Lutwak-Yang-Zhang in 2016. Variational formulas are…

We characterize the symmetric measures which satisfy the one dimensional convex Poincar\'e inequality. For these measures the tenesorization argument yields concentration inequalities for their products and convex sets in R^n.

In this paper, the $q$-th dual curvature measure is extended to convex functions and the associated Minkowski problem is posed. A special case includes the $q$-th dual curvature measure of convex bodies which defined by Huang, Lutwak, Yang…

We establish concentration inequalities in the class of ultra log-concave distributions. In particular, we show that ultra log-concave distributions satisfy Poisson concentration bounds. As an application, we derive concentration bounds for…

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.

We prove several stability and volume difference inequalities for projections of convex bodies and apply them to prove a hyperplane inequality for surface area of projection bodies.

We prove pinching estimates for dual flows provided the curvature function used in the inverse flow in de Sitter space is convex.