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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.
![$ k $](/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png)
![$ d $](/files/tex/aeba4a4076fc495e8b5df04d874f2911a838883a.png)
![$ \ge d $](/files/tex/616a25c014115485d010b8d180f072cc22ed1c53.png)
![$ k $](/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png)
![$ k $](/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png)
Continous analogue of Hirsch conjecture ★★
Author(s): Deza; Terlaky; Zinchenko
Conjecture The order of the largest total curvature of the primal central path over all polytopes defined by
inequalities in dimension
is
.
![$ n $](/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png)
![$ d $](/files/tex/aeba4a4076fc495e8b5df04d874f2911a838883a.png)
![$ n $](/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png)
Average diameter of a bounded cell of a simple arrangement ★★
Author(s): Deza; Terlaky; Zinchenko
Conjecture The average diameter of a bounded cell of a simple arrangement defined by
hyperplanes in dimension
is not greater than
.
![$ n $](/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png)
![$ d $](/files/tex/aeba4a4076fc495e8b5df04d874f2911a838883a.png)
![$ d $](/files/tex/aeba4a4076fc495e8b5df04d874f2911a838883a.png)
Keywords: arrangement; diameter; polytope
Fat 4-polytopes ★★★
Author(s): Eppstein; Kuperberg; Ziegler
The fatness of a 4-polytope is defined to be
where
is the number of faces of
of dimension
.
Question Does there exist a fixed constant
so that every convex 4-polytope has fatness at most
?
![$ c $](/files/tex/dccee841f3f498c2c58fa6ae1c1403c5a88c5b8d.png)
![$ c $](/files/tex/dccee841f3f498c2c58fa6ae1c1403c5a88c5b8d.png)
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