polytope

Geometry » Polytopes

Durer's Conjecture ★★★

Author(s): Durer; Shephard

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

Keywords: folding; polytope

Posted by dmoskovich
updated June 4th, 2012

add new comment

Geometry » Polytopes

Cube-Simplex conjecture ★★★

Author(s): Kalai

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

Keywords: cube; facet; polytope; simplex

Posted by mdevos
updated May 9th, 2008

add new comment

Geometry » Polytopes

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 $n$ inequalities in dimension $d$ is $n$ .

Keywords: curvature; polytope

Posted by deza
updated September 30th, 2007

add new comment

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 $n$ hyperplanes in dimension $d$ is not greater than $d$ .

Keywords: arrangement; diameter; polytope

Posted by deza
updated September 30th, 2007

add new comment

Geometry » Polytopes

Fat 4-polytopes ★★★

Author(s): Eppstein; Kuperberg; Ziegler

The fatness of a 4-polytope $P$ is defined to be $(f_1 + f_2)/(f_0 + f_3)$ where $f_i$ is the number of faces of $P$ of dimension $i$ .

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

Keywords: f-vector; polytope

Posted by mdevos
updated May 12th, 2007

add new comment

Geometry » Polytopes

Hirsch Conjecture ★★★

Author(s): Hirsch

Conjecture Let $P$ be a convex $d$ -polytope with $n$ facets. Then the diameter of the graph of the polytope $P$ is at most $n-d$ .

Keywords: diameter; polytope

Posted by Robert Samal
updated July 9th, 2011