THE D-STEP CONJECTURE FOR POLYHEDRA OF DIMENSION D<6,
Abstract
Two functions A and B, of interest in combinatorial geometry and the theory of linear programming, are defined and studied. A(d,n) is the maximum diameter of convex polyhedra of dimension d with n faces of dimension d-1; similarly, B(d,n) is the maximum diameter of bounded polyhedra of dimension d with n faces of dimension d-1. The diameter of a polyhedron P is the smallest integer k such that any two vertices of P can be joined by a path of k or fewer edges of P. It is shown that the bounded d-step conjecture, i.e. B(d,2d) = d, is true for d < or 5. It is also shown that the general d-step conjecture, i.e. A(d,2d) < or = d, of significance in linear programming, is false for d equal to or greater than 4. A number of other specific values and bounds for A and B are presented.
Document Details
- Document Type
- Technical Report
- Publication Date
- Dec 01, 1965
- Accession Number
- AD0628199
Entities
People
- David W. Walkup
- Victor Klee
Organizations
- Boeing