Related papers: An inequality for length and volume in the complex…
The long-standing Godbersen's conjecture asserts that the Rogers-Shephard inequality for the volume of the difference body is refined by an inequality for the mixed volume of a convex body and its reflection about the origin. The conjecture…
In this short survey we want to present some of the impact of Minkowski's successive minima within Convex and Discrete Geometry. Originally related to the volume of an $o$-symmetric convex body, we point out relations of the successive…
On the class of log-concave functions on $\R^n$, endowed with a suitable algebraic structure, we study the first variation of the total mass functional, which corresponds to the volume of convex bodies when restricted to the subclass of…
We prove a bound for the geodesic diameter of a subset of the unit ball in $\mathbb{R}^n$ described by a fixed number of quadratic equations and inequalities, which is polynomial in $n$, whereas the known bound for general degree is…
It has recently been established byWang and Xia [WX] that local minimizers of perimeter within a ball subject to a volume constraint must be spherical caps or planes through the origin. This verifies a conjecture of the authors and is in…
We establish central limit theorems for natural volumes of random inscribed polytopes in projective Riemannian or Finsler geometries. In addition, normal approximation of dual volumes and the mean width of random polyhedral sets are…
This article belongs to the area of geometric tomography, which is the study of geometric properties of solids based on data about their sections and projections. We describe a new direction in geometric tomography where different…
The main result of this paper is an inequality relating the lattice point enumerator of a 3-dimensional, 0-symmetric convex body and its successive minima. This is an example of generalization of Minkowski's theorems on successive minima,…
In his paper "On the Schlafli differential equality", J. Milnor conjectured that the volume of n-dimensional hyperbolic and spherical simplices, as a function of the dihedral angles, extends continuously to the closure of the space of…
We propose a new notion of singularity in General Relativity which complements the usual notions of geodesic incompleteness and curvature singularities. Concretely, we say that a spacetime has a volume singularity if there exist points…
We investigate the properties of convex functions in the plane that satisfy a local inequality which generalizes the notion of sub-solution of Monge-Ampere equation for a Monge-Kantorovich problem with quadratic cost between non-absolutely…
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.
Geodesics on the infinite dimensional symmetric space $\hcal$ of K\"ahler metrics in a fixed K\"ahler class on a projective K\"ahler manifold X are solutions of a homogeneous complex Monge-Amp\`ere equation in $X \times A$, where $A \subset…
We construct a class of Finsler metrics in three-dimensional space such that all their geodesics are lines, but not all planes are extremal for their Hausdorff area functionals. This shows that if the Hausdorff measure is used as notion of…
We consider the volume-constrained minimization of the sum of the perimeter and the Riesz potential. We add an external potential of the form $\|x\|^{\beta}$ that provides the existence of a minimizer for any volume constraint, and we study…
We establish volume comparison results for balls in Riemannian manifolds with $C^{1,1}$-metrics with a lower bound on the Ricci tensor and for the evolution of spacelike, acausal, causally complete hypersurfaces with an upper bound on the…
Every closed geodesic $\gamma$ on a surface has a canonically associated knot $\widehat\gamma$ in the projective unit tangent bundle. We study, for $\gamma$ filling, the volume of the associated knot complement with respect to its unique…
We focus on the analysis of local minimizers of the Mahler volume, that is to say the local solutions to the problem $$\min\{ M(K):=|K||K^\circ|\;/\;K\subset\R^d\textrm{ open and convex}, K=-K\}, $$ where $K^\circ:=\{\xi\in\R^d ; \forall…
We prove a relative volume comparison theorem of Petersen-Wei for both $L^P$-bound of Bakry-\'{E}mery Ricci curvature and gradient of potential function. As an application, we give a modified proof for a volume comparison and monotonicity…
In this note we show the following result using the integral-geometric formula of R. Howard: Consider the totally geodesic $\mathbb{R}P^{2m}$ in $\mathbb{C}P^n$. Then it minimizes volume among the isotropic submanifolds in the same…