Category Archives: Convex polytopes

Beyond the g-conjecture – algebraic combinatorics of cellular spaces I

The g-conjecture for spheres is surely the one single conjecture I worked on more than on any other, and also here on the blog we had a sequence of posts about it by Eran Nevo (I,II,III,IV). Here is a great … Continue reading

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A Mysterious Duality Relation for 4-dimensional Polytopes.

Two dimensions Before we talk about 4 dimensions let us recall some basic facts about 2 dimensions: A planar polygon has the same number of vertices and edges. This fact, which just asserts that the Euler characteristic of is zero, … Continue reading

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My Copy of Branko Grünbaum’s Convex Polytopes

Branko Grünbaum is my academic grandfather (see this highly entertaining post for a picture representing five academic generations). Gunter Ziegler just wrote a beautiful article in the Notices of the AMS on Branko Grunbaum’s  classic book “Convex Polytopes”, so this … Continue reading

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Eran Nevo: g-conjecture part 4, Generalizations and Special Cases

This is the fourth in a series of posts by Eran Nevo on the g-conjecture. Eran’s first post was devoted to the combinatorics of the g-conjecture and was followed by a further post by me on the origin of the g-conjecture. Eran’s second post was about … Continue reading

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Touching Simplices and Polytopes: Perles’ argument

Joseph Zaks (1984), picture taken by Ludwig Danzer (OberWolfach photo collection)   The story I am going to tell here was told in several places, but it might be new to some readers and I will mention my own angle, … Continue reading

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Convex Polytopes: Seperation, Expansion, Chordality, and Approximations of Smooth Bodies

I am happy to report on two beautiful results on convex polytopes. One disproves an old conjecture of mine and one proves an old conjecture of mine. Loiskekoski and Ziegler: Simple polytopes without small separators. Does Lipton-Tarjan’s theorem extends to high … Continue reading

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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

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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

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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

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