Related papers: A Proof of the Simplex Mean Width Conjecture

Given a convex set and an interior point close to the boundary, we prove the existence of a supporting hyperplane whose distance to the point is controlled, in a dimensionally quantified way, by the thickness of the convex set in the…

P\'al's classical isominwidth inequality states that the regular triangle has minimal area among plane convex bodies of minimal width $w$. A similar result is the Blaschke--Lebesgue inequality that states that Reuleaux triangles minimize…

The oloid is the convex hull of two circles with equal radius in perpendicular planes so that the center of each circle lies on the other circle. We calculate the mean width of the oloid in two ways, first via the integral of mean…

In this note we link symplectic and convex geometry by relating two seemingly different open conjectures: a symplectic isoperimetric-type inequality for convex domains, and Mahler's conjecture on the volume product of centrally symmetric…

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…

A variant of the flatness problem from integer programming is studied, in which one considers convex bodies in $\mathbb{R}^d$ with at most $k$ interior lattice points. The maximum lattice width of such a body is denoted by Flt(d,k) and it…

For a convex body $K\subset\mathbb{R}^d$ the mean distance $\Delta(K)=\mathbb{E}|X_1-X_2|$ is the expected Euclidean distance of two independent and uniformly distributed random points $X_1,X_2\in K$. Optimal lower and upper bounds for…

For any origin-symmetric convex body $K$ in $\mathbb{R}^n$ in isotropic position, we obtain the bound: \[ M^*(K) \leq C \sqrt{n} \log(n)^2 L_K ~, \] where $M^*(K)$ denotes (half) the mean-width of $K$, $L_K$ is the isotropic constant of…

This thesis consists of five papers about reduced spherical convex bodies and in particular spherical bodies of constant width on the $d$-dimensional sphere $S^d$. In paper I we present some facts describing the shape of reduced bodies of…

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…

The width of a convex curve in the plane is the minimal distance between a pair of parallel supporting lines of the curve. In this paper we study the width of nodal lines of eigenfunctions of the Laplacian on the standard flat torus. We…

A centrally symmetric convex body is a convex compact set with non-empty interior that is symmetric about the origin. Of particular interest are those that are both smooth and strictly convex -- known here as regular symmetric bodies --…

Similarly to the classic notion in $E^d$, a subset of a positive diameter below $\frac{\pi}{2}$ of a hemisphere of the sphere $S^d$ is called complete, provided adding any extra point increases its diameter. Complete sets are convex bodies…

The variance conjecture in Asymptotic Convex Geometry stipulates that the Euclidean norm of a random vector uniformly distributed in a (properly normalised) high-dimensional convex body $K\subset {\mathbb R}^n$ satisfies a Poincar\'e-type…

We show that the number of lattice directions in which a d-dimensional convex body in R^d has minimum width is at most 3^d-1, with equality only for the regular cross-polytope. This is deduced from a sharpened version of the 3^d-theorem due…

We obtain a sharp characterization of the Euclidean ball among all convex bodies K whose boundary has a pointwise k-th mean curvature not smaller than a geometric constant at almost all normal points. This geometric constant depends only on…

High proved the following theorem. If the intersections of any two congruent copies of a plane convex body are centrally symmetric, then this body is a circle. In our paper we extend the theorem of High to spherical, Euclidean and…

We study the regularity of solutions to an optimal transportation problem where the dimension of the source is larger than that of the target. We demonstrate that if the target is $c$-convex, then the source has a canonical foliation whose…

In this work we discuss a conjecture of Viterbo relating the symplectic capacity of a convex body and its volume. The conjecture states that among all 2n-dimensional convex bodies with a given volume the euclidean ball has maximal…

Let $M$ be a compact $n$-dimensional Riemannian manifold with nonnegative Ricci curvature and mean convex boundary $\partial M$. Assume that the mean curvature $H$ of the boundary $\partial M$ satisfies $H \geq (n-1) k >0$ for some positive…