Publication

Comments on: Recent progress on the combinatorial diameter of polytopes and simplicial complexes

Friedrich Eisenbrand
2013
Journal Articles

Abstract

This paper features two main contributions. On the one hand, it gives an impressive survey on the progress on the diameter problem, including the breakthrough of the author with his disproof of the Hirsch conjecture among many other recent results. On the other hand, it features new results (exponential lower bounds) on the diameter of simplicial complexes and re-interprets the recent activities of the polymath 3 project that has been coordinated by Gil Kalai.

Official source

https://infoscience.epfl.ch/entities/publication/82beb70d-be57-43b2-af64-a78bb23b7d90

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Friedrich Eisenbrand
2013
Journal Articles

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/82beb70d-be57-43b2-af64-a78bb23b7d90

About this result

Related concepts (17)

Polytope

In elementary geometry, a polytope is a geometric object with flat sides (faces). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that the sides of a (k + 1)-polytope consist of k-polytopes that may have (k – 1)-polytopes in common.

Regular polytope

In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces (for all 0 ≤ j ≤ n, where n is the dimension of the polytope) — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of dimension ≤ n. Regular polytopes are the generalized analog in any number of dimensions of regular polygons (for example, the square or the regular pentagon) and regular polyhedra (for example, the cube).

4-polytope

In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), and cells (polyhedra). Each face is shared by exactly two cells. The 4-polytopes were discovered by the Swiss mathematician Ludwig Schläfli before 1853. The two-dimensional analogue of a 4-polytope is a polygon, and the three-dimensional analogue is a polyhedron.

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