Abstract We prove that for any semi-norm $\|\cdot \|$ on $\mathbb{R}^n$ and symmetric convex body $K$ in $\mathbb{R}^n,$(1)$$\begin{align}& \int_{\partial K} \frac{\|n_x\|^2}{\langle x,n_x\rangle}\leq \frac{1}{|K|}\left(\int_{\partial \|n_x\| \right)^2, \end{align}$$and characterize the equality cases of this new inequality. The above would also follow from Log-Brunn–Minkowski conjecture if...

Abstract. For metric measure spaces verifying the reduced curvature-dimension condition CD∗(K,N) we prove a series of sharp functional inequalities under the additional assumption of essentially nonbranching. Examples of spaces entering this framework are (weighted) Riemannian manifolds satisfying lower Ricci curvature bounds and their measured Gromov Hausdorff limits, Alexandrov spaces satisfy...

We strengthen some known stability results from the Brunn-Minkowski theory and obtain new results of similar types. These results concern pairs of convex bodies for which either surface area measures, or counterparts of such measures in the Brunn-Minkowski-Firey theory, or geometrically significant transforms of such measures, are close to each other. MSC 2000: 52A20, 52A40.

In this paper, we firstly generalize the Brunn-Minkowski type inequality for Ekeland-Hofer-Zehnder symplectic capacity of bounded convex domains established by Artstein-Avidan-Ostrover in 2008 to extended capacities constructed authors based on a class Hamiltonian non-periodic boundary value problems recently. Then introduce billiards domains, and them prove some corresponding results those per...

We initiate a systematic investigation into the nature of the function αK (L, ρ) that gives the volume of the intersection of one convex body K in Rn and a dilatate ρL of another convex body L in Rn, as well as the function ηK (L, ρ) that gives the (n − 1)-dimensional Hausdorff measure of the intersection of K and the boundary ∂(ρL) of ρL. The focus is on the concavity properties of αK (L, ρ). ...

This note is devoted to the study of the dependence on p of the constant in the reverse Brunn-Minkowski inequality for p-convex balls (i.e. p-convex symmetric bodies). We will show that this constant is estimated as c ≤ C(p) ≤ C , for absolute constants c > 1 and C > 1. Let K ⊂ IR n and 0 < p ≤ 1. K is called a p-convex set if for any λ, μ ∈ (0, 1) such that λ + μ = 1 and for any points x, y ∈ ...