Finding Optimal Shadows of Polytopes

Deformed Products and Maximal Shadows of Polytopes

We present a construction of deformed products of polytopes that has as special cases all the known constructions of linear programs with \many pivots," starting with the famous Klee-Minty cubes from 1972. Thus we obtain sharp estimates for the following geometric quantities for d-dimensional simple polytopes with at most n facets: the maximal number of vertices on an increasing path, the maxim...

Finding Shadows among Disks

Given a set of n non-overlapping unit disks in the plane, a line ` is called blocked if it intersects at least one of the disks and a point p is called a shadow point if all lines containing p are blocked. In addition, a maximal closed set of shadow points is called a shadow region. We derive properties of shadow regions, and present an O(n) algorithm that outputs all shadow regions. We prove t...

On Computing the Shadows and Slices of Polytopes

We study the projection of polytopes along k orthogonal vectors for various input and output forms. We show that if k is part of the input and we are interested in output-sensitive algorithms, then in most forms the problem is equivalent to enumerating vertices of polytopes, except in two where it is NP-hard. In two other forms the problem is trivial. We also review the complexity of computing ...

Complexity of Finding Nearest Colorful Polytopes

Let P1, . . . , Pd+1 ⊂ R be point sets whose convex hulls each contain the origin. Each set represents a color class. The Colorful Carathéodory theorem guarantees the existence of a colorful choice, i.e., a set that contains exactly one point from each color class, whose convex hull also contains the origin. The computational complexity of finding such a colorful choice is still unknown. We stu...

Using shadows in finding surface orientations