Gromov’s angular spread theorem

Karim AdiprasitoLeave a comment

I recently reported about a polyhedral proof of Stoker’s theorem using a lemma of Martin Winter; this, in turn, uses technique by Izmestiev.

I wanted to take a brief moment to illustrate that technique; Gromov, a few years ago, published a marvelous little paper. He used differential geometry techniques to show that a polytope cannot be too long in every direction. Specifically: Consider P a d-dimensional polytope, and map it cellularly to a d-cube C. Pick two opposing facets F_-, F_+ of that d-cube. Their preimages in P are d-1-disks.

Nice enough. Now, try to go from the preimage of F_- to the preimage of F_+. If you go from one facet to an adjacent one in P, then you “pay” the angle between their normals. All together, the optimal way you can take gives a minimal angular distance between the preimage of F_- to the preimage of F_+ in P; it describes how far the two preimages are apart.

Now, it would be nice, and perhaps point towards the polynomial Hirsch conjecture, if we could say something about this distance. But Gromov says: geometry naturally gives something different: If I minimize over ALL the opposing pairs of facets of C, then I obtain \square_{\rangle}^{\,n}(P), the angular spread… it describes the question whether P is long in every direction. Gromov’s paper obtains a displayed O(n^{3/2}) bound by passing through smooth mean-curvature geometry [1]. He needs to smoothen the polytope first, and his argument for non-simple polytopes is a bit … fishy. In comes Izmestiev, and I want to illustrate his method here, and improve the bound using it.

Misha
Ivan
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IDP polytopes, hypersimplices, projective spaces

Karim AdiprasitoLeave a comment

I wanted to update on 3 small new results.

First, Luis Ferroni used LLM (and a lot of persistence and cleverness) to prove something I long suspected: that IDP lattice polytopes do NOT have unimodal h∗h^\ast vectors. Unimodality is nice because it makes the sequence especially simple: it rises, then falls again… a dromedary, instead of a camel. And that IDP polytopes (that is, polytopes which are generated by atoms on the same level, that is, level one) satisfy it was a (I thought unlikely, but long standing) conjecture of Stanley….

Luis found, quite amazingly, that these are rather nice polytopes: smooth Cayley polytopes of rectangular prisms. This complements our work on IDP polytopes, proving among other things monotonicity of the h∗h^\ast vector in the second half. Congratulations Luis!

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Misha, again

Karim AdiprasitoLeave a comment

On 26 of August this year, Misha Verbitsky was fired from his position at IMPA for his blog.

I do not agree with many of his posts, and some verge on the barely tolerable for me. But I would also respect his opinion, discussion. Most of all, I think his opinion has no bearing on his mathematics.

University leadership and institutional actors may believe themselves immune to scrutiny. In this particular case, I want to highlight that merely two months ago, the same person that now fired him wrote an article about defending specifically his freedom of speech. Please share. I also attach the letter regarding his dismissal. It appears that Misha was not presented with an actual accusation he could answer, and that the posts he was (supposedly) dismissed for kept changing as the institute adapted its narrative, suggesting post hoc justification.

https://www1.folha.uol.com.br/colunas/marceloviana/2026/06/misha-verbitsky-e-a-importancia-da-liberdade-de-expressao.shtml

Stoker, again

Karim AdiprasitoLeave a comment

I wanted to quickly mention an “almost” LLM result. I wrote previously about a marvelous new approach of Yuchen Bi to Stoker’s conjecture using differential geometry, but that still critically used quite extensively non-polyhedral techniques. Yesterday, Arseniy Akopyan wrote to me that he chatted with LLM and found a polyhedral only proof.

Not essentially new, Arseniy and I then noticed that the LLM had “stolen” a result of Martin Winter, who in 2023 found the proof without noticing, and wrote it in a very nice paper expanding our knowledge of Wachspress coordinates and Izmestiev matrices. Congratulations, Martin!

Martin Winter
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A little update on Stoker’s conjecture

Karim AdiprasitoLeave a comment

Recovering from some surgeries, I wanted to write a little something on Stoker’s conjecture: It is best viewed as an angular converse to Cauchy’s rigidity theorem. Cauchy’s proves more than the usual statement that congruent corresponding faces determine a convex polyhedron: for a fixed combinatorial type, the angles inside the faces already determine the dihedral angles. In 1968, James J. Stoker asked whether the converse holds: do the dihedral angles determine the face angles?

Eleazer Bromberg, James J. Stoker, and Louis Nirenberg, ca 1960

(Ps: For health reasons, I am taking a break and may not check or respond to communication)

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Misha Verbitsky facing extradition to Russia

Karim AdiprasitoLeave a comment

Misha Verbitsky, a distinguished mathematician and long-time public critic of the Russian state, has reportedly been detained at Yerevan airport on a Russian request.

Verbitsky is known not only for his mathematical work, but also for his uncompromising public writing: against war, against censorship, in favor of open culture and free expression. One need not agree with everything he has written to see the danger here. The Russian charges against him arise in the context of political speech and dissent. Extradition to Russia would therefore place him at grave risk.

Armenia should not hand him over. At minimum, Verbitsky must have immediate access to lawyers, independent observers, and a fair procedure in which the political nature of the Russian request is taken seriously.

This is urgent. Please circulate reliable information, contact academic and human-rights networks, and ask Armenian authorities not to extradite Misha Verbitsky to Russia.

For questions, please contact his daughter, Sima

see also change.org/free_misha_verbitsky

Update: he was freed and left Armenia

Singularities from bug to feature: Homeomorphisms over K, neighborhoods and waists

Karim AdiprasitoLeave a comment

There is nothing new in this post. It is the sort of thing that most people would probably call folklore or obvious, but then again, sometimes it is cool to write it down. And who knows, perhaps someone finds it useful. Also… let’s face it, a lot of PL/combinatorial topology is a mess, and things that used to be well known get lost.

There are, on occasion, obvious concepts which lack a name. Perhaps because they have not proven useful yet. But then again, they are so very natural, they deserve to be discussed somewhere.

If X and Y are two topological spaces, perhaps manifolds, perhaps something else, then one remembers from topology what it means for them to be homeomorphic. We also know what it means to be PL homeomorphic1, that is when the homeomorphism isn’t merely continuous, but piecewise linear. Still, it pays sometime to go back in time and look at the history of the subject. Incidentally, let me mention this marvelous book on the Hauptvermutung and its history I found recently by the late Yuli Rudyak2.

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Hauptvermutung, Tietze’s dream and the importance of checking your references

Karim Adiprasito2 Comments

Recently, we got back several referee reports for this paper here (to appear in Inventiones mathematicae, see here) which were quite positive (well, the comments were “the construction is genius and it works, but the exposition is quite messy”; I choose to take it as a compliment), which made me very happy because it is a very classical conjecture with some nice implications. And seeing people who took their time to understand the construction (and wade through my horrible writing, because the “messy” is certainly my fault) made me quite proud…

Speaking of, though, classical conjecture…. well one part was Oda’s. That was fine. The other conjecture we solved was “Alexander’s conjecture”, named after James Waddell Alexander. Everyone kept calling it that, and Alexander certainly worked on related problems and proved something weaker in a 1930 paper (building on work of Max Newman, who, another cool fact, is best known for becoming a codebreaker later).

James Waddell Alexander
Max Newman
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