Related papers: An algorithm to reconstruct convex polyhedra from …
Let K be a convex body in $R^d$. A random polytope is the convex hull $[x_1,...,x_n]$ of finitely many points chosen at random in K. $\Bbb E(K,n)$ is the expectation of the volume of a random polytope of n randomly chosen points. I.…
Estimating the volume of a convex body is a canonical problem in theoretical computer science. Its study has led to major advances in randomized algorithms, Markov chain theory, and computational geometry. In particular, determining the…
We are interested in the fast computation of the exact value of integrals of polynomial functions over convex polyhedra. We present speed ups and extensions of the algorithms presented in previous work. We present the new software…
The intrinsic volumes of a convex cone are geometric functionals that return basic structural information about the cone. Recent research has demonstrated that conic intrinsic volumes are valuable for understanding the behavior of random…
In this paper, motivated by the work of Edelman and Strang, we show that for fixed integers $d\geq 2$ and $n\geq d+1$ the configuration space of all facet volume vectors of all $d$-polytopes in $\mathbb R^{d}$ with $n$ facets is a full…
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…
We briefly introduce several problems: (1) a generalization of the convex fair partition conjecture, (2) on non-trivial invariants among polyhedrons that can be formed from the same set of face polygons, (3) two questions on assembling…
In many cases the convexity of the image of a linear map with range is $R^n$ is automatic because of the facial structure of the domain of the map. We develop a four step procedure for proving this kind of ``automatic convexity''. To make…
Binary tomography is concerned with the recovery of binary images from a few of their projections (i.e., sums of the pixel values along various directions). To reconstruct an image from noisy projection data, one can pose it as a…
Given a polyhedral surface, assume that it is prohibited to change the shape and size of any face but it is permissible to change the dihedral angles between the faces. A polyhedral surface is said to be flexible if it is possible to change…
In a previous work we proved that each $n$-dimensional convex polyhedron ${\mathcal K}subset{\mathbb R}^n$ and its relative interior are regular images of ${\mathbb R}^n$. As the image of a non-constant polynomial map is an unbounded…
A convex polyhedron $P$ is $k$-equiprojective if all of its orthogonal projections, i.e., shadows, except those parallel to the faces of $P$ are $k$-gon for some fixed value of $k$. Since 1968, it is an open problem to construct all…
In this paper we consider a class of right-angled polyhedra in three-dimensional Lobachevsky space, all vertices of which lie on the absolute. New upper bounds on volumes in terms the number of faces of the polyhedron are obtained. Volumes…
We study the two-dimensional geometric knapsack problem for convex polygons. Given a set of weighted convex polygons and a square knapsack, the goal is to select the most profitable subset of the given polygons that fits non-overlappingly…
We develop mask iterative hard thresholding algorithms (mask IHT and mask DORE) for sparse image reconstruction of objects with known contour. The measurements follow a noisy underdetermined linear model common in the compressive sampling…
We construct a quasi-polynomial time deterministic approximation algorithm for computing the volume of an independent set polytope with restrictions. Randomized polynomial time approximation algorithms for computing the volume of a convex…
We present a geometric algorithm to compute the geometric kernel of a generic polyhedron. The geometric kernel (or simply kernel) is definedas the set of points from which the whole polyhedron is visible. Whilst the computation of the…
Generalized permutahedra are the polytopes obtained from the permutahedron by changing the edge lengths while preserving the edge directions, possibly identifying vertices along the way. We introduce a "lifting" construction for these…
Decomposition of shapes into (approximate) convex parts is essential for applications such as part-based shape representation, shape matching, and collision detection. In this paper, we propose a novel convex decomposition using a…
Volume approximation is an important problem found in many applications of computer graphics, vision, and image processing. The problem is about computing an accurate and compact approximate representation of 3D volumes using some simple…