Durer's Conjecture ★★★

Author(s): Durer; Shephard

\begin{conjecture} Every convex polytope has a non-overlapping edge unfolding. \end{conjecture}

Keywords: folding; polytope

Cube-Simplex conjecture ★★★

Author(s): Kalai

\begin{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. \end{conjecture}

Keywords: cube; facet; polytope; simplex

Continous analogue of Hirsch conjecture ★★

Author(s): Deza; Terlaky; Zinchenko

\begin{conjecture} The order of the largest total curvature of the primal central path over all polytopes defined by $n$ inequalities in dimension $d$ is $n$. \end{conjecture}

Keywords: curvature; polytope

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

Author(s): Deza; Terlaky; Zinchenko

\begin{conjecture} The average diameter of a bounded cell of a simple arrangement defined by $n$ hyperplanes in dimension $d$ is not greater than $d$. \end{conjecture}

Keywords: arrangement; diameter; polytope

Fat 4-polytopes ★★★

Author(s): Eppstein; Kuperberg; Ziegler

The \emph{fatness} of a 4-\Def{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 $i$.

\begin{question} Does there exist a fixed constant $c$ so that every convex 4-polytope has fatness at most $c$? \end{question}

Keywords: f-vector; polytope

Hirsch Conjecture ★★★

Author(s): Hirsch

\begin{conjecture} Let $P$ be a convex $d$-\Def{polytope} with $n$ \Def[facets]{Facet_(mathematics)}. Then the diameter of the graph of the polytope $P$ is at most $n-d$. \end{conjecture}

Keywords: diameter; polytope

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