# polytope

## Durer's Conjecture ★★★

**Conjecture**Every convex polytope has a non-overlapping edge unfolding.

## Cube-Simplex conjecture ★★★

Author(s): Kalai

**Conjecture**For every positive integer , there exists an integer so that every polytope of dimension has a -dimensional face which is either a simplex or is combinatorially isomorphic to a -dimensional cube.

## Continous analogue of Hirsch conjecture ★★

Author(s): Deza; Terlaky; Zinchenko

**Conjecture**The order of the largest total curvature of the primal central path over all polytopes defined by inequalities in dimension is .

## Average diameter of a bounded cell of a simple arrangement ★★

Author(s): Deza; Terlaky; Zinchenko

**Conjecture**The average diameter of a bounded cell of a simple arrangement defined by hyperplanes in dimension is not greater than .

Keywords: arrangement; diameter; polytope

## Fat 4-polytopes ★★★

Author(s): Eppstein; Kuperberg; Ziegler

The *fatness* of a 4-polytope is defined to be where is the number of faces of of dimension .

**Question**Does there exist a fixed constant so that every convex 4-polytope has fatness at most ?