Related papers: Cone-volume measure and stability

In 1960, Gr\"{u}nbaum proved that for any convex body $C\subset\mathbb{R}^d$ and every halfspace $H$ containing the centroid of $C$, one has that the volume of $H\cap C$ is at least a $\frac{1}{e}$-fraction of the volume of $C$. Recently,…

The mixed Christoffel problem asks for necessary and sufficient conditions for a Borel measure on the Euclidean unit sphere to be the mixed area measure of some convex bodies, all but one of them are fixed. We consider the case in which the…

In this paper, extending the work of Gal'perin (Comm. Math. Phys. 154: 63-84, 1993), we investigate generalizations of the concepts of centroids and static equilibrium points of a convex body in spherical, hyperbolic and normed spaces. In…

This paper introduces a natural definition for the volume of the unit ball in $n$-dimensional normed spaces $\mathbb{R}^n$. This definition preserves the Euclidean relation $P(B)/V(B)=n$ between the perimiter and the volume of the unit ball…

It is well known that any measure in S^2 satisfying certain simple conditions is the surface measure of a bounded convex body in R^3. It is also known that a local perturbation of the surface measure may lead to a nonlocal perturbation of…

Let $K \subset {\mathbb R}^2$ be an $o$-symmetric convex body, and $K^*$ its polar body. Then we have $|K|\cdot |K^*| \ge 8$, with equality if and only if $K$ is a parallelogram. ($| \cdot |$ denotes volume). If $K \subset {\mathbb R}^2$ is…

Equivalent conditions that make the normal cone maximal monotone are investigated in the general settings of locally convex spaces. Some consequences such as Bishop Phelps and sum representability results are presented in the last part.

In this paper, we prove Mahler's conjecture concerning the volume product of centrally symmetric convex bodies in $\mathbb{R}^n$ in the case where $n=3$. Furthermore, we determine the equality condition.

We study some geometric properties of the $L_q$-centroid bodies $Z_q(\mu)$ of an isotropic log-concave measure $\mu $ on ${\mathbb R}^n$. For any $2\ls q\ls\sqrt{n}$ and for $\varepsilon \in (\varepsilon_0(q,n),1)$ we determine the inradius…

In this paper, we prove the stability of metric measure spaces satisfying the curvature-dimension condition for negative dimensions under a concentration topology. This result is an analogue of the result by Funano-Shioya with respect to…

We prove a randomized version of the generalized Urysohn inequality relating mean-width to the other intrinsic volumes. To do this, we introduce a stochastic approximation procedure that sees each convex body K as the limit of intersections…

We establish the log-concavity of the volume of central sections of dilations of the cross-polytope (the strong B-inequality for the cross-polytope and Lebesgue measure restricted to an arbitrary subspace).

We consider convex sets whose modulus of convexity is uniformly quadratic. First, we observe several interesting relations between different positions of such ``2-convex'' bodies; in particular, the isotropic position is a finite…

We develop a technique using dual mixed-volumes to study the isotropic constants of some classes of spaces. In particular, we recover, strengthen and generalize results of Ball and Junge concerning the isotropic constants of subspaces and…

We investigate a potential obtained as the convolution of a radially symmetric function and the characteristic function of a body (the closure of a bonded open set) with exterior cones. In order to restrict the location of a maximizer of…

This paper addresses the three concepts of \textit{ consistency, stability and convergence } in the context of compact finite volume schemes for systems of nonlinear hyperbolic conservation laws. The treatment utilizes the framework of…

We apply the convection stability criterion to a fluid in global thermodynamic equilibrium with a rigid rotation or with a constant acceleration along the streamlines. Different equations of state describing strongly interacting matter are…

Given the significance of physical measures in understanding the complexity of dynamical systems as well as the noisy nature of real-world systems, investigating the stability of physical measures under noise perturbations is undoubtedly a…

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…

We show that in all dimensions d>2, there exists an asymmetric convex body of revolution all of whose maximal hyperplane sections have the same volume. This gives the negative answer to the question posed by V. Klee in 1969.