## Periodic Reporting for period 2 - PolSymAGA (Polarity and Central-Symmetry in Asymptotic Geometric Analysis)

Reporting period: 2020-03-01 to 2021-08-31

In the project we address 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) serve as motivation to understand more deeply these problems and their connections to other related topics.

As these are problems within basic research in mathematics, their importance for society is not immediately apparent, but as science has shown many times, advances in basic research lead to applications in related, more applicable scientific fields.

For example, in one part of the project we studied the polarity transform, a cousin of the widely applied Legendre 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.

Several long standing open problems (Mahler's conjecture, Viterbo's conjecture, Godbersen's conjecture) serve as motivation to understand more deeply these problems and their connections to other related topics.

As these are problems within basic research in mathematics, their importance for society is not immediately apparent, but as science has shown many times, advances in basic research lead to applications in related, more applicable scientific fields.

For example, in one part of the project we studied the polarity transform, a cousin of the widely applied Legendre 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 the direction of symmetry measures: 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. Other variations are also of high interest; a new and exciting position called "saddle John" is introduced; it's extremal properties understood as well as the associated measures. In a work with Sadovsky and Sanyal on questions of symmetry we apply combintorial 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.

In the direction of billiards: A work with Florentin, Ostrover and Rosen where we study caustics, prove that duality exists in the euclidean case but does not hold in general. Joint work with Chor in work in progress has related results on symplectic John-position. There is an exciting decomposition of identity associated with this position, where instead of contact points we have contact pairs in symplectic positions.

In direction of polarity: A work with Florentin and Segal on the polar Prekopa Leindler inequality, and the derived functional inequality of the form of polar Poincare, polar Log Sobolev and Polar Brascamp and Lieb. These seems to be in very close connection with out joint work with Barel in her 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. 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 the next step which is measure concentration, entropy inequalities and log-sobolev type inequalities for polarity (and as stated, these also emanate form our polar Prekopa Leindelr inequality, so weaving the two together is our next goal).

Also in the direction of polarity, a new family of transforms has come up, some one-parameter perturbation of the classical ones, and these give rise to new characterization results.

In direction of covering, joint work with Slomka has been published on the functional covering number, and ongoing work with Roysdon and with Falah (a new MSc student) seem to lead to deeper connections between covering numbers, isotropicity and some form of a Banach Mazur space for log concave functions. 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 think about the topology and convergence in these spaces.

In the direction of billiards: A work with Florentin, Ostrover and Rosen where we study caustics, prove that duality exists in the euclidean case but does not hold in general. Joint work with Chor in work in progress has related results on symplectic John-position. There is an exciting decomposition of identity associated with this position, where instead of contact points we have contact pairs in symplectic positions.

In direction of polarity: A work with Florentin and Segal on the polar Prekopa Leindler inequality, and the derived functional inequality of the form of polar Poincare, polar Log Sobolev and Polar Brascamp and Lieb. These seems to be in very close connection with out joint work with Barel in her 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. 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 the next step which is measure concentration, entropy inequalities and log-sobolev type inequalities for polarity (and as stated, these also emanate form our polar Prekopa Leindelr inequality, so weaving the two together is our next goal).

Also in the direction of polarity, a new family of transforms has come up, some one-parameter perturbation of the classical ones, and these give rise to new characterization results.

In direction of covering, joint work with Slomka has been published on the functional covering number, and ongoing work with Roysdon and with Falah (a new MSc student) seem to lead to deeper connections between covering numbers, isotropicity and some form of a Banach Mazur space for log concave functions. 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 think about the topology and convergence in these spaces.

The plans for the remaining part of the project are:

Completing and applying the Brenier-type results following from out transportation theory.

This will influence the polarity and t-polarity project, including Concentration, Log-Sobolev and entropy.

The work of Mussnig and Li gives a new angle yet to be fully understood on the metrics and topologies on some spaces of functions involved.

Work with Roysdon where the p-summation replaces the usual ones will hopefully lead to new links.

The project of c-duality instead of polarity is leading us towards new c-duality transforms, and their implications can be tremendous.

Capacity progress with Arnon Chor leads to new symplectcic versions of Brascamp-Lieb inequality and new John-type decomposition, which give us a new lead towards Viterbo's conjecture (and isoperimetric inequality for capacities)

The work with Putterman is still not complete and seems to lead to new positions with better properties.

Completing and applying the Brenier-type results following from out transportation theory.

This will influence the polarity and t-polarity project, including Concentration, Log-Sobolev and entropy.

The work of Mussnig and Li gives a new angle yet to be fully understood on the metrics and topologies on some spaces of functions involved.

Work with Roysdon where the p-summation replaces the usual ones will hopefully lead to new links.

The project of c-duality instead of polarity is leading us towards new c-duality transforms, and their implications can be tremendous.

Capacity progress with Arnon Chor leads to new symplectcic versions of Brascamp-Lieb inequality and new John-type decomposition, which give us a new lead towards Viterbo's conjecture (and isoperimetric inequality for capacities)

The work with Putterman is still not complete and seems to lead to new positions with better properties.