Viterbo’s conjecture was refuted by Pazit Haim-Kislev and Yaron Ostrover

Viterbo conjecture – refuted

Claude Viterbo’s 2000 volume-capacity conjecture asserts that the Euclidean (even dimensional) ball maximizes  (every) symplectic capacity  among convex bodies of the same volume. In the recent paper A Counterexample to Viterbo’s Conjecture, Pazit Haim-Kislev and Yaron Ostrover disproved the conjecture. 

Pazit Haim-Kislev and Yaron Ostrover

We discussed some background on symplectic geometry in this post.  Gromov’s non-squeezing theorem(1985) asserts that a ball of radius R cannot be “squeezed” by a symplectic map to a cylinder whose base is a circle of radius r,  for R > r. 

Gromov’s theorem can be seen as the starting point of a rich theory of symplectic “capacities”.  The main property of a symplectic capacity is that it does not decrease under symplectic embeddings. The theorem immediately led to two definitions of symplectic capacity for one of which the assertion of Viterbo’s conjecture is obvious while for the other it was unknown. The Gromov width of is the largest radius (in fact, the supremum of radii) of a ball that can be symplectically embedded in .  The cylindrical capacity is the infimum over all radii of cylinders that can be embedded into.   

Viterbo further conjectured that all symplectic capacities coincide on convex bodies of dimension . It was known that several symplectic capacities do coincide for convex bodies.  The paper studied the Ekeland–Hofer–Zehnder (EHZ) capacity which refers to different capacities that coincide for convex sets. Their value is related to important properties of billiards; more precisely, Minkowski billiards introduced by Ugene Gutkin and Sergei Tabachnikov. Let me remark that my convexity fellows Dániel Bedzek and Károly Bedzek found a useful way to compute billiard trajectories and, believe it or not, Helly’s theorem plays a role there.  

Remark: Even after our semester long Kazhdan seminar (with Dorit, Leonid, and Guy) on symplectic geometry and quantum computing my intuitions about symplectic capacities are very poor. Experts warned me that symplectic capacities are very different from volume-based measurements. (For example, in the axiomatic definition of symplectic capacities, the “symplectic size” of the infinite cylinder is – in particular, it is finite!) 

The Example

The example is a direct product of two pentagons.

Computing symplectic capacities of polytopes

In the paper On the symplectic size of convex polytopes Pazit Haim-Kislev found a combinatorial formula to calculate the symplectic capacity (EHZ) of polytopes (this was her M. Sc. thesis), and in her home page she provides an implementation.

In addition to this computer computation the new paper by Pazit and Yaron provides a short “human” proof for the capacity of the example. 

The connection to Mahler’s conjecture

Some connections to convex geometry were mentioned in this post. The Mahler conjecture asserts that for every centrally symmetric body

Here is the polar dual of . Shiri Artstein-Avidan, Roman Karasev and Yaron Ostrover found in 2013 a remarkable connection to the Mahler’s conjecture:

Theorem (Artstein-Avidan, Karasev, and Ostrover): For centrally symmetric convex bodies the Mahler conjecture is equivalent to the special case of Viterbo conjecture for convex bodies of the form , where is a centrally symmetric convex body. 

An early paper on from 2006 on the connection between convex geometry and symplectic geometry was written by Shiri Artstein-Avidan, Vitali Milman, and Yaron Ostrover. Here is its short abstract:

In this work we bring together tools and ideology from two different fields, Symplectic Geometry and Asymptotic Geometric Analysis, to arrive at some new results. Our main result is a dimension-independent bound for the symplectic capacity of a convex body by its volume radius.

The connection to the 3ᵈ conjecture and the flag conjecture.

Let be a centrally-symmetric convex polytope and let be its barycentric subdivisions. I conjectured that attains its minimum for the -cube. Two special cases for are:

The 3ᵈ-conjecture: Let P be a centrally symmetric d-polytope. Then P has at least 3ᵈ non-empty faces.

The flag conjecture: Let P be a centrally symmetric d-polytope. Then P has at least 2ᵈd! saturated flag of faces. 

Equality for the Mahler conjecture, The 3ᵈ-conjecture, and the flag conjecture are attained for a large class of polytopes called Hanner polytopes (and perhaps only for them). For more on these conjectures see this post.

Dmitry Faifman, Constantin Vernicos, and Cormac Walsh proposed in their paper Volume growth of Funk geometry and the flags of polytopes an exciting conjecture that interpolates between Mahler’s conjecture and my flag conjecture. I hope to come back to this conjecture in a future post.

Could it be that all these conjectures are wrong?

Experts hope (and I certainly share these hopes) that Viterbo’s conjecture holds for centrally symmetric convex domains (as well as Mahler’s conjecture, the , and the Flag conjecture). But we have to prepare for other possibilities as well.