Related papers: Algebraic Distance Optimization in Polyhedral Norm…
The Hilbert metric is a distance function defined for points lying within a convex body. It generalizes the Cayley-Klein model of hyperbolic geometry to any convex set, and it has numerous applications in the analysis and processing of…
It is well known that the solution of topology optimization problems may be affected both by the geometric properties of the computational mesh, which can steer the minimization process towards local (and non-physical) minima, and by the…
Although recovering an Euclidean distance matrix from noisy observations is a common problem in practice, how well this could be done remains largely unknown. To fill in this void, we study a simple distance matrix estimate based upon the…
We consider offsets of a union of convex objects. We aim for a filtration, a sequence of nested cell complexes, that captures the topological evolution of the offsets for increasing radii. We describe methods to compute a filtration based…
In this paper, we first consider a flat plate (called a lamina) with uniform density $\rho$ that occupies a region $\mathfrak R$ of the plane. We show that the location of the center of mass, also known as the centroid, of the region equals…
Consider a polyhedral convex cone which is given by a finite number of linear inequalities. We investigate the problem to project this cone into a subspace and show that this problem is closely related to linear vector optimization: We…
This is an experimental case study in real algebraic geometry, aimed at computing the image of a semialgebraic subset of 3-space under a polynomial map into the plane. For general instances, the boundary of the image is given by two highly…
We develop a moduli theory of algebraic varieties and pairs of non-negative Kodaira dimension. We define stable minimal models and construct their projective coarse moduli spaces under certain natural conditions. This can be applied to a…
Let $\Gamma$ denote a $Q$-polynomial distance-regular graph with diameter $D\geq 1$. For a vertex $x$ of $\Gamma$ the corresponding subconstituent algebra $T=T(x)$ is generated by the adjacency matrix $A$ of $\Gamma$ and the dual adjacency…
We study the computational complexity of determining the Hausdorff distance of two polytopes given in halfspace- or vertex-presentation in arbitrary dimension. Subsequently, a matching problem is investigated where a convex body is allowed…
Let $\Xi$ be a discrete set in ${\mathbb{R}}^d$. Call the elements of $\Xi$ centers. The well-known Voronoi tessellation partitions ${\mathbb{R}}^d$ into polyhedral regions (of varying sizes) by allocating each site of ${\mathbb{R}}^d$ to…
We propose a new method for simplifying semidefinite programs (SDP) inspired by symmetry reduction. Specifically, we show if an orthogonal projection map satisfies certain invariance conditions, restricting to its range yields an equivalent…
Constructing a discretization of a given set is a major problem in various theoretical and applied disciplines. An offset discretization of a set $X$ is obtained by taking the integer points inside a closed neighborhood of $X$ of a certain…
Some basic mathematical tools such as convex sets, polytopes and combinatorial topology, are used quite heavily in applied fields such as geometric modeling, meshing, computer vision, medical imaging and robotics. This report may be viewed…
Given an input matrix polynomial whose coefficients are floating point numbers, we consider the problem of finding the nearest matrix polynomial which has rank at most a specified value. This generalizes the problem of finding a nearest…
We study the problem of partitioning a polygon into the minimum number of subpolygons using cuts in predetermined directions such that each resulting subpolygon satisfies a given width constraint. A polygon satisfies the unit-width…
We consider the Voronoi diagram of points in the real plane when the distance between two points $a$ and $b$ is given by $L_p(a-b)$ where $L_p((x,y)) = (|x|^p+|y|^p)^{1/p}.$ We prove that the Voronoi diagram has a limit as $p$ converges to…
We generalize the family of $(\sigma, \rho)$-problems and locally checkable vertex partition problems to their distance versions, which naturally captures well-known problems such as distance-$r$ dominating set and distance-$r$ independent…
We investigate numerical differentiation formulas on irregular centers in two or more variables that are exact for polynomials of a given order and minimize an absolute seminorm of the weight vector. Error bounds are given in terms of a…
In this article we define an algebraic vertex of a generalized polyhedron and show that it is the smallest set of points needed to define the polyhedron. We prove that the indicator function of a generalized polytope $P$ is a linear…