http://www.ifor.math.ethz.ch/~fukuda/polyfaq/polyfaq.html

**Komei Fukuda
Swiss Federal Institute of Technology
Lausanne and Zurich, Switzerland
fukuda@ifor.math.ethz.ch
**

*Date:* Version June 18, 2004

- Contents
- What is Polyhedral Computation FAQ?
- Convex Polyhedron
- What is convex polytope/polyhedron?
- What are the faces of a convex polytope/polyhedron?
- What is the face lattice of a convex polytope
- What is a dual of a convex polytope?
- What is simplex?
- What is cube/hypercube/cross polytope?
- What is simple/simplicial polytope?
- What is 0-1 polytope?
- What is the best upper bound of the numbers of -dimensional faces of a -polytope with vertices?
- What is convex hull? What is the convex hull problem?
- What is the Minkowski-Weyl theorem for convex polyhedra?
- What is the vertex enumeration problem, and what is the facet enumeration problem?
- How can one enumerate all faces of a convex polyhedron?
- What computer models are appropriate for the polyhedral computation?
- How do we measure the complexity of a convex hull algorithm?
- How many facets does the average polytope with vertices in have?
- How many facets can a 0-1 polytope with vertices in have?
- How hard is it to verify that an H-polyhedron and a V-polyhedron are equal?
- Is there an efficient way of determining whether a given point is in the convex hull of a given finite set of points in ?
- How can one remove all interior points of from for large clouds of points in ?
- Is there any efficient algorithm to remove redundant inequalities from a system of linear inequalities
- Is there any efficient algorithm to compute the intersection of two (or ) polytopes
- Is there any efficient algorithm to compute the volume of a convex polytope in ?

- Voronoi Diagram and Delaunay Triangulation
- What is cell complex? What is triangulation?
- What is Voronoi diagram in ?
- What is the Delaunay triangulation in ?
- Computing the Delaunay complex and the Voronoi diagram. What does it mean and how to do it with available software?
- Is it possible to compute only the adjacencies of Voronoi cells in the Voronoi diagram efficiently?
- Is it possible to compute only the edges of the Delaunay complex (triangulation) ?
- Is it possible to determine the Delaunay cell containing a given point efficiently?
- What is the best upper bound of the numbers of simplices in the Delaunay triangulation?

- Linear Programming

- Polyhedral Computation Codes
- Acknowledgements
- Bibliography
- About this document ...

Komei Fukuda 2004-08-26