# Category Archives: Convex polytopes

## The Simplex, the Cyclic polytope, the Positroidron, the Amplituhedron, and Beyond

A quick schematic road-map to these new geometric objects. The  positroidron can be seen as a cellular structure on the nonnegative Grassmanian – the part of the real Grassmanian G(m,n) which corresponds to m by n matrices with all m by … Continue reading

## My Mathematical Dialogue with Jürgen Eckhoff

Jürgen Eckhoff, Ascona 1999 Jürgen Eckhoff is a German mathematician working in the areas of convexity and combinatorics. Our mathematical paths have met a remarkable number of times. We also met quite a few times in person since our first … Continue reading

Posted in Combinatorics, Convex polytopes, Open problems | | 1 Comment

## Many triangulated three-spheres!

The news Eran Nevo and Stedman Wilson have constructed triangulations with n vertices of the 3-dimensional sphere! This settled an old problem which stood open for several decades. Here is a link to their paper How many n-vertex triangulations does the 3 … Continue reading

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## Richard Stanley: How the Proof of the Upper Bound Theorem (for spheres) was Found

The upper bound theorem asserts that among all d-dimensional polytopes with n vertices, the cyclic polytope maximizes the number of facets (and k-faces for every k). It was proved by McMullen for polytopes in 1970, and by Stanley for general triangulations … Continue reading

Posted in Combinatorics, Convex polytopes | | 3 Comments

## F ≤ 4E

1. E ≤ 3V Let G be a simple planar graph with V vertices and E edges. It follows from Euler’s theorem that E ≤ 3V In fact, we have (when V is at least 3,) that E ≤ 3V – 6. … Continue reading

## Lionel Pournin found a combinatorial proof for Sleator-Tarjan-Thurston diameter result

I just saw in Claire Mathieu’s blog  “A CS professor blog” that a simple proof of the Sleator-Tarjan-Thurston’s diameter result for the graph of the associahedron was found by Lionel Pournin! Here are slides of his lecture “The diameters of associahedra” … Continue reading

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## Karim Adiprasito: Flag simplicial complexes and the non-revisiting path conjecture (A combinatorial proof of the Adiprasito-Benedetti theorem.)

This post is authored by Karim Adiprasito The past months have seen some exciting progress on diameter bounds for polytopes and polytopal complexes, both in the negative and in the positive direction.  Jesus de Loera and Steve Klee described simplicial polytopes which are not  … Continue reading

## Tokyo, Kyoto, and Nagoya

Near Nagoya: Firework festival; Kyoto: with Gunter Ziegler; with Takayuki Hibi, Hibi, Marge Bayer, Curtis Green and Richard Stanly; Tokyo: Peter Frankl; crowded crossing. Added later: Mazi and I at the same restaurant taken by Stanley. I just returned from … Continue reading

Posted in Combinatorics, Conferences, Convex polytopes | | 2 Comments

## Satoshi Murai and Eran Nevo proved the Generalized Lower Bound Conjecture.

Satoshi Murai and Eran Nevo have just proved the 1971 generalized lower bound conjecture of McMullen and Walkup, in their  paper On the generalized lower bound conjecture for polytopes and spheres . Let me tell you a little about it. … Continue reading

## Updates, Boolean Functions Conference, and a Surprising Application to Polytope Theory

The Debate continues The debate between Aram Harrow and me on Godel Lost letter and P=NP (GLL) regarding quantum fault tolerance continues. The first post entitled Perpetual  motions of the 21th century featured mainly my work, with a short response by Aram. … Continue reading