Related papers: A Note on the Size of the Largest Ball Inside a Co…

Let $\boldsymbol{\tau}=(\tau_1,\dots,\tau_m)\in \mathbb{R}_{\ge 0}^m$ satisfy $\sum_{i=1}^m \tau_i>1$ and $\tau_1\ge \cdots \ge \tau_m$ Let $\Psi_{\boldsymbol\tau}=(\psi_1,\dots,\psi_m)$ be given by $$ \psi_i(q)=q^{-\tau_i}, \qquad…

The polygon $P$ is small if its diameter equals one. When $n=2^s$, it is still an open problem to find the maximum perimeter or the maximum width of a small $n$-gon. Motivated by Bingane's series of works, we improve the lower bounds for…

The classic double bubble theorem says that the least-perimeter way to enclose and separate two prescribed volumes in $\mathbb{R}^N$ is the standard double bubble. We seek the optimal double bubble in $\mathbb{R}^N$ with density, which we…

In this paper we study the problem of maximizing the distance to a given point over an intersection of balls. It was already known that this problem can be solved in polynomial time and space if the given point is not in the convex hull of…

Consider a random set of points on the unit sphere in $\mathbb{R}^d$, which can be either uniformly sampled or a Poisson point process. Its convex hull is a random inscribed polytope, whose boundary approximates the sphere. We focus on the…

The maximum volume $j$-simplex problem asks to compute the $j$-dimensional simplex of maximum volume inside the convex hull of a given set of $n$ points in $\mathbb{Q}^d$. We give a deterministic approximation algorithm for this problem…

We consider a compact hyperbolic antiprism. It is a convex polyhedron with $2n$ vertices in the hyperbolic space $\mathbb{H}^3$. This polyhedron has a symmetry group $S_{2n}$ generated by a mirror-rotational symmetry of order $2n$, i.e.…

Suppose $<A_i, \vec{c}_i>$ are planar (convex) H-polyhedra, that is, $A_i \in \mathbb{R}^{n_i \times 2}$ and $\vec{c}_i \in \mathbb{R}^{n_i}$. Let $P_i = \{\vec{x} \in \mathbb{R}^2 \mid A_i\vec{x} \leq \vec{c}_i \}$ and $n = n_1 + n_2$. We…

Let $X_1,\ldots,X_n$ be independent random points in the closed unit ball of $\mathbb{R}^d$. Assume that each $X_i$ has a beta distribution with parameter $\beta_i \ge -1$: if $\beta_i>-1$, then $X_i$ has Lebesgue density proportional to…

We establish a general formula for the maximum size of finite length block codes with minimum pairwise distance no less than $d$. The achievability argument involves an iterative construction of a set of radius-$d$ balls, each centered at a…

Let n >= 2 be an integer and consider the set T_n of n by n permutation matrices pi for which pi_{ij}=0 for j>=i+2. In this paper we study the convex hull of T_n, which we denote by P_n. P_n is a polytope of dimension binom{n}{2}. Our main…

We introduce the M-representation of polytopes, which makes it possible to compute linear transformations, convex hulls, and Minkowski sums with linear complexity in the dimension of the polytopes. When the polytope is a convex hull of a…

We explore an instance of the question of partitioning a polygon into pieces, each of which is as ``circular'' as possible, in the sense of having an aspect ratio close to 1. The aspect ratio of a polygon is the ratio of the diameters of…

We study the smallest intersecting and enclosing ball problems in Euclidean spaces for input objects that are compact and convex. They link and unify many problems in computational geometry and machine learning. We show that both problems…

We consider the class of $\lambda$-concave bodies in $\mathbb R^{n+1}$; that is, convex bodies with the property that each of their boundary points supports a tangent ball of radius $1/\lambda$ that lies locally (around the boundary point)…

A body moves in a medium composed of noninteracting point particles; interaction of particles with the body is absolutely elastic. It is required to find the body's shape minimizing or maximizing resistance of the medium to its motion. This…

We propose a cut-based algorithm for finding all vertices and all facets of the convex hull of all integer points of a polyhedron defined by a system of linear inequalities. Our algorithm DDM Cuts is based on the Gomory cuts and the dynamic…

Given an undirected graph $G = (V,E)$, the cut polytope $\mathrm{CUT}(G)$ is defined as the convex hull of the incidence vectors of all cuts in $G$. The 1-skeleton of $\mathrm{CUT}(G)$ is a graph whose vertex set is the vertex set of the…

We consider the problem of estimating the distance between two bodies of volume $\varepsilon$ located inside a $n$-dimensional ball $U$ of unit volume for $n\to\infty$. Let $A$ be a closed set with a smooth boundary of the volume…

In this work we approach the problem of determining which (compact) semialgebraic subsets of ${\mathbb R}^n$ are images under polynomial maps $f:{\mathbb R}^m\to{\mathbb R}^n$ of the closed unit ball $\overline{{\mathcal B}}_m$ centered at…