On the volume of the double stochastic matrices.
On the ψ₂-behaviour of linear functionals on isotropic convex bodies
The slicing problem can be reduced to the study of isotropic convex bodies K with , where is the isotropic constant. We study the ψ₂-behaviour of linear functionals on this class of bodies. It is proved that for all θ in a subset U of with measure σ(U) ≥ 1 - exp(-c√n). However, there exist isotropic convex bodies K with uniformly bounded geometric distance from the Euclidean ball, such that . In a different direction, we show that good average ψ₂-behaviour of linear functionals on an isotropic...
On Tucker's key theorem.
On two classes of Banach spaces with uniform normal structure
On Two Conjectures of Franz Hering About Convex Surfaces.
On Two Finite Covering Problems of Bambah, Rogers, Woods and Zassenhaus.
On unit balls and isoperimetrices in normed spaces
The purpose of this paper is to continue the investigations on the homothety of unit balls and isoperimetrices in higher-dimensional Minkowski spaces for the Holmes-Thompson measure and the Busemann measure. Moreover, we show a strong relation between affine isoperimetric inequalities and Minkowski geometry by proving some new related inequalities.
On Weak Tail Domination of Random Vectors
Motivated by a question of Krzysztof Oleszkiewicz we study a notion of weak tail domination of random vectors. We show that if the dominating random variable is sufficiently regular then weak tail domination implies strong tail domination. In particular, a positive answer to Oleszkiewicz's question would follow from the so-called Bernoulli conjecture. We also prove that any unconditional logarithmically concave distribution is strongly dominated by a product symmetric exponential measure.
On weighted parallel volumes.
On α(·)-monotone multifunctions and differentiability of γ-paraconvex functions
Let (X,d) be a metric space. Let Φ be a family of real-valued functions defined on X. Sufficient conditions are given for an α(·)-monotone multifunction to be single-valued and continuous on a weakly angle-small set. As an application it is shown that a γ-paraconvex function defined on an open convex subset of a Banach space having separable dual is Fréchet differentiable on a residual set.
One functional-analytical idea by Alexandrov in convex geometry.
Only ellipsoids have caustics.
Operations between sets in geometry
An investigation is launched into the fundamental characteristics of operations on and between sets, with a focus on compact convex sets and star sets (compact sets star-shaped with respect to the origin) in -dimensional Euclidean space . It is proved that if , with three trivial exceptions, an operation between origin-symmetric compact convex sets is continuous in the Hausdorff metric, covariant, and associative if and only if it is addition for some . It is also demonstrated that if ,...
Optimal bounds for the colored Tverberg problem
We prove a “Tverberg type” multiple intersection theorem. It strengthens the prime case of the original Tverberg theorem from 1966, as well as the topological Tverberg theorem of Bárány et al. (1980), by adding color constraints. It also provides an improved bound for the (topological) colored Tverberg problem of Bárány & Larman (1992) that is tight in the prime case and asymptotically optimal in the general case. The proof is based on relative equivariant obstruction theory.
Optimal isometries for a pair of compact convex subsets of ℝⁿ
In 1989 R. Arnold proved that for every pair (A,B) of compact convex subsets of ℝ there is an Euclidean isometry optimal with respect to L₂ metric and if f₀ is such an isometry, then the Steiner points of f₀(A) and B coincide. In the present paper we solve related problems for metrics topologically equivalent to the Hausdorff metric, in particular for metrics for all p ≥ 2 and the symmetric difference metric.
Optimality conditions for maximizers of the information divergence from an exponential family
The information divergence of a probability measure from an exponential family over a finite set is defined as infimum of the divergences of from subject to . All directional derivatives of the divergence from are explicitly found. To this end, behaviour of the conjugate of a log-Laplace transform on the boundary of its domain is analysed. The first order conditions for to be a maximizer of the divergence from are presented, including new ones when is not projectable to .
Optimierung konstant belegter Ovale und ein Hyperbel-Schnittsatz
Optimizing geometric measures for fixed minimal annulus and inradius.
Orthant spanning simplexes with minimal volume.
Outer -convex functions on a normed space.