Related papers: On random disc-polygons in a disc-polygon
Let $f \in \mathbb{Z}[y]$ be a polynomial such that $f(\mathbb{N}) \subseteq \mathbb{N}$, and let $p_{\mathcal{A}_{f}}(n)$ denote number of partitions of $n$ whose parts lie in the set $\mathcal{A}_f:=\{f(n):n \in \mathbb{N}\}$. Under…
The goal of this paper is to establish certain inequalities between the numbers of convex polytopes in the d-dimensional space "containing" and "avoiding" zero provided that their vertex sets are subsets of a given finite set of points in…
Let $U_1,\ldots,U_n$ be independent random vectors uniformly distributed on the unit sphere $\mathbb S^{d-1}\subseteq\mathbb R^d$, where $n\ge d$, and consider the random polyhedral cone \[ \mathcal W_{n,d}:=\mathop{\mathrm{pos}}…
Pick $n$ points $Z_0,...,Z_{n-1}$ uniformly and independently at random in a compact convex set $H$ with non empty interior of the plane, and let $Q^n_H$ be the probability that the $Z_i$'s are the vertices of a convex polygon. Blaschke…
Existence of nicely bounded sections of two symmetric convex bodies K and L implies that the intersection of random rotations of K and L is nicely bounded. For L = subspace, this main result immediately yields the unexpected phenomenon: "If…
We present a probabilistic model of illuminating a convex body by independently distributed light sources. In addition to recovering C.A. Rogers' upper bounds for the illumination number, we improve previous estimates of J. Januszewski and…
The main objective of this article is to provide an alternative approach to the central result of [Eldred, A. Anthony, Kirk, W. A., Veeramani, P., Proximal normal structure and relatively nonexpansive mappings, Studia Math., vol 171(3),…
We study asymptotic clustering of zeros of random polynomials, and show that the expected discrepancy of roots of a polynomial of degree $n$, with not necessarily independent coefficients, decays like $\sqrt{\log n/n}$. Our proofs rely on…
We establish, for a generically big Hermitian line bundle, the convergence of truncated Harder-Narasimhan polygons and the uniform continuity of the limit. As applications, we prove a conjecture of Moriwaki asserting that the arithmetic…
We determine upper asymptotic estimates of Kolmogorov and linear $n$-widths of unit balls in Sobolev and Besov norms in $L_{p}$-spaces on smooth compact Riemannian manifolds. For compact homogeneous manifolds, we establish estimates which…
As a natural analog of Urysohn's inequality in Euclidean space, Gao, Hug, and Schneider showed in 2003 that in spherical or hyperbolic space, the total measure of totally geodesic hypersurfaces meeting a given convex body K is minimized…
In this article we consider non-convex $4d$ polytopes in $\mathbb{R}^4$. The paper consist of two parts: Firstly, we extend the proof of the formula for the $4d$ volume in terms of $2d$ face bivectors and boundary graph crossings from…
Computing mixed volume of convex polytopes is an important problem in computational algebraic geometry. This paper establishes sufficient conditions under which the mixed volume of several convex polytopes exactly equals the normalized…
Given a set of disjoint simple polygons $\sigma_1, \ldots, \sigma_n$, of total complexity $N$, consider a convexification process that repeatedly replaces a polygon by its convex hull, and any two (by now convex) polygons that intersect by…
We consider the set of points chosen randomly, independently and uniformly in the $d$-dimensional spherical layer. A set of points is called $1$-convex if all its points are vertices of the convex hull of this set. In \cite{3} an estimate…
Many asymptotic formulas exist for unrestricted integer partitions as well as for distinct partitions of integers into a finite number of parts. Szekeres and Canfield have derived an asymptotic formula for the number of partitions that is…
Enumeration of various types of lattice polygons and in particular polyominoes is of primary importance in many machine learning, pattern recognition, and geometric analysis problems. In this work, we develop a large deviation principle for…
Extending classical results on polytopal approximation of convex bodies, we derive asymptotic formulas for the weighted approximation of smooth convex functions by piecewise affine convex functions as the number of their facets tends to…
Every convex body K in R^n has a coordinate projection PK that contains at least vol(0.1 K) cells of the integer lattice PZ^n, provided this volume is at least one. Our proof of this counterpart of Minkowski's theorem is based on an…
We establish a general criterion for the existence of convex sets of fixed shape as, e.g., balls of a given radius, of maximal probability on Banach spaces. We also provide counterexamples showing that their existence my fail even in some…