Periodic Reporting for period 5 - PolSymAGA (Polarity and Central-Symmetry in Asymptotic Geometric Analysis)
Berichtszeitraum: 2024-09-01 bis 2025-08-31
In the project we addressed several topics in asymptotic convex geometry, mainly connected with polarity transform, symmetry measures and symplectic aspects. Several long standing open problems (Mahler's conjecture, Viterbo's conjecture, Godbersen's conjecture) served as motivation to understand more deeply these problems and their connections to other related topics.
As science has shown many times, advances in basic research lead to applications in related, more applicable scientific fields. For example, in studying the polarity transform, we found out it is induced by a special cost function, which is sometimes infinite valued. It turns out several key ingredients in optimal transport theory fail for such untraditional costs, and so we had to invent the relevant theory. This new theory is now applicable to many different cost functions in various situations, and helps prove in an elegant and natural way many old results with long technical proofs. This exemplifies how new problems shed light and help build techniques which are then applicable in a very wide context.
As science has shown many times, advances in basic research lead to applications in related, more applicable scientific fields. For example, in studying the polarity transform, we found out it is induced by a special cost function, which is sometimes infinite valued. It turns out several key ingredients in optimal transport theory fail for such untraditional costs, and so we had to invent the relevant theory. This new theory is now applicable to many different cost functions in various situations, and helps prove in an elegant and natural way many old results with long technical proofs. This exemplifies how new problems shed light and help build techniques which are then applicable in a very wide context.
In a work with Putterman we complemented previous work on maximal intersection position to apply to any pair of bodies; the intuition obtained from various differentiation techniques allowed Putterman to prove that a local log BM inequality implies the global one, using a very simple geometric idea. We introduced the saddle-John position of one body inside another, proving that it shares some of the properties possessed by the position of maximal volume, and can be used to improve volume ratio estimates. We also discuss the maximal intersection position of one body with respect to another, and show the existence of a natural decomposition of identity associated to this position, extending previous work. In a work with Sadovsky and Sanyal on questions of symmetry we apply combinatorial intuition to prove results for anti-blocking bodies, which give us an almost-Mahler inequality for Cayley bodies, as well as several cases of Godbersen's conjecture. Shay then proved the more general case of locally anti-blocking. This family of bodies turned out to be quite important, and recently Arnon Chor proved Kalai's flag conjecture was shown for these (ArXiV).
In a work with Florentin, Ostrover and Rosen we study caustics, prove that duality exists in the euclidean case but does not hold in general. Meanwhile Ostrover and Haim-Kislev found a counterexample to Viterbo's conjecture in the non-symmetric case. In a work with Florentin and Segal we gave a polar Prekopa Leindler inequality, and derived functional inequalities of the form of polar Poincare, Log Sobolev andBrascamp and Lieb. These are connected to Barel's MSc thesis, and its continuation with Sadovsky and Wyczesany, where we study the Wasserstein distance with respect to a general cost. To this end we found a Rockafellar type theorem applicable for non traditional cost, where cyclic monotonicity is replaced by the stronger condition of c-path boundedness together with "no black holes". Using this condition, one may show that two strongly c-compatible measures admit a transport plan supported on a c-subgradient. This leads to a deeper understanding of the underlying structure, paving the way to measure concentration, entropy inequalities and log-sobolev type inequalities for polarity. The Hall-polytope plays a role in the case of discrete measures, producing an alternative combinatorial proof. A new family of transforms has come up, a one-parameter perturbation of the classical ones, and these give rise to new characterization results (under preparation). In addition, questions of general c-duality which arose from the abstract settings in which we preformed the Rockafellar-type theorems gave rise to a "zoo of set dualities" each of which is a rich world with many connections to different geometric problems. In "zoo of dualities” we also provided a new Gaussian Blaschke-Santalo inequality, as well as some completely new directions such as pseudo-cones duality and ball bodies duality. The ball-bodies duality project continued in two recent papers published, demonstrating rigidity of duality in this new class on the one hand, and the exciting fact that it is actually an isometry, and the unique one. This in turn inspired a definition by Schutt Werner and Yalikun of a ball-floating body and ball-type surface area. This opened up a huge family of questions which we are currently addressing in several sub-projects.
In a paper with Putterman we give new bounds for the mixed volume of a body and it minus, giving new bounds for Godbersen's problem.
Joint work with Slomka a paper has been published on the functional covering number, and with Falah we investigated boundary-type Brunn-Minkowski ineqaulities. This in turn inspired a study of vertex generated polytopes, which in particular produce a parameter which quantifies the improvement of Maurey-Lemma covering bound for these classes of polytopes. We show many results regarding this class, among them: that the class contains all zonotopes, that it is dense in dimension 2, that any polytope can be summed with a zonotope so that the sum is in this class. We introduce for every polytope a parameter which measures how far it is from being vertex-generated, and show that when this parameter is small, strong covering properties hold. The study geometry of log-concave function has been further put forward in the paper of Li and Mussnig where they link several different ways to define topology and convergence in these spaces. Mussnig continued with many Hadwiger type functional results, joint with coauthors.
A summary of the duality direction was given in my ECM talk in Seville 2024 and the paper of the proceedings of this event, and another summary I am submitting now to the ICM 2026 proceedings, and I will speak about it in my talk at ICM 2026 in Philadelphia.
In a work with Florentin, Ostrover and Rosen we study caustics, prove that duality exists in the euclidean case but does not hold in general. Meanwhile Ostrover and Haim-Kislev found a counterexample to Viterbo's conjecture in the non-symmetric case. In a work with Florentin and Segal we gave a polar Prekopa Leindler inequality, and derived functional inequalities of the form of polar Poincare, Log Sobolev andBrascamp and Lieb. These are connected to Barel's MSc thesis, and its continuation with Sadovsky and Wyczesany, where we study the Wasserstein distance with respect to a general cost. To this end we found a Rockafellar type theorem applicable for non traditional cost, where cyclic monotonicity is replaced by the stronger condition of c-path boundedness together with "no black holes". Using this condition, one may show that two strongly c-compatible measures admit a transport plan supported on a c-subgradient. This leads to a deeper understanding of the underlying structure, paving the way to measure concentration, entropy inequalities and log-sobolev type inequalities for polarity. The Hall-polytope plays a role in the case of discrete measures, producing an alternative combinatorial proof. A new family of transforms has come up, a one-parameter perturbation of the classical ones, and these give rise to new characterization results (under preparation). In addition, questions of general c-duality which arose from the abstract settings in which we preformed the Rockafellar-type theorems gave rise to a "zoo of set dualities" each of which is a rich world with many connections to different geometric problems. In "zoo of dualities” we also provided a new Gaussian Blaschke-Santalo inequality, as well as some completely new directions such as pseudo-cones duality and ball bodies duality. The ball-bodies duality project continued in two recent papers published, demonstrating rigidity of duality in this new class on the one hand, and the exciting fact that it is actually an isometry, and the unique one. This in turn inspired a definition by Schutt Werner and Yalikun of a ball-floating body and ball-type surface area. This opened up a huge family of questions which we are currently addressing in several sub-projects.
In a paper with Putterman we give new bounds for the mixed volume of a body and it minus, giving new bounds for Godbersen's problem.
Joint work with Slomka a paper has been published on the functional covering number, and with Falah we investigated boundary-type Brunn-Minkowski ineqaulities. This in turn inspired a study of vertex generated polytopes, which in particular produce a parameter which quantifies the improvement of Maurey-Lemma covering bound for these classes of polytopes. We show many results regarding this class, among them: that the class contains all zonotopes, that it is dense in dimension 2, that any polytope can be summed with a zonotope so that the sum is in this class. We introduce for every polytope a parameter which measures how far it is from being vertex-generated, and show that when this parameter is small, strong covering properties hold. The study geometry of log-concave function has been further put forward in the paper of Li and Mussnig where they link several different ways to define topology and convergence in these spaces. Mussnig continued with many Hadwiger type functional results, joint with coauthors.
A summary of the duality direction was given in my ECM talk in Seville 2024 and the paper of the proceedings of this event, and another summary I am submitting now to the ICM 2026 proceedings, and I will speak about it in my talk at ICM 2026 in Philadelphia.
With the end of the grants duration, all of the directions have been significantly advanced. As is usually the case, this means that there are now even more directions to explore, since the more we understand we can highlight new goals and try to pass new boundaries. This is definitely the case for this project, and the research plans for the next few years are extensive. From figuring out the role of the new dualities we have discovered, to fully understanding the measure transportation with respect to new non-traditional costs, through new classes of bodies such as vertex generated or ball bodies, what functional and geometric inequalities they satisfy, with extending applications in analysis, geometry and combinatorics.