Optimal transport is a powerful framework, used by mathematicians and economists, that studies the allocation of resources. The origin of the subject goes back to the engineer Gaspard Monge, who was interested in the problem of the optimal way of redistributing mass: given a pile of soil, how can it be transported and reshaped to form an embankment with minimal effort? Originally, the cost function of transportation was simply the distance the particles were moved, but as the world evolved, scientists became interested in other, often abstract, cost functions. 

In the well-developed classical theory of transportation, one considers finite-valued cost functions (such as the distance function). However, sometimes one may want to prohibit specific transportation schemes, which would correspond to setting the cost function to +∞. Despite the interest in this important case, it was only recently resolved by Shiri Artstein-Avidan (Tel Aviv University), Shay Sadovsky (Tel Aviv University), and Kasia Wyczesany (Tel Aviv University). See [2, 3]. 

When some pairs of points cannot be mapped to one another, one needs to impose certain compatibility conditions so that a finite-cost transport scheme even exists. This compatibility between transported measures is expressed using ‘cost duality’. It turns out that this notion is not only useful when studying transportation but is of independent interest in geometry. Indeed, in the recent follow-up paper A Zoo of Dualities [1], the authors study cost dualities and show that any order-reversing quasi-involution on subsets can be represented as a cost duality. This gives a general characterization of order-reversing quasi-involutions that unifies our understanding and exposes the underlying principles and structures. 

The cost representation of all order-reversing quasi-involutions motivates a search for general results. For example, the authors study invariant sets of order-reversing quasi-involutions, i.e. fixed points of the transform, which often have extremal properties within a given family of sets with respect to some functional. This can be observed in the Blaschke-Santaló inequality, which asserts that among centrally symmetric convex bodies K in Rⁿ the volume product, defined as Vol(K)Vol(K°), is maximized by the unique invariant set of polarity transform °, which is the Euclidean unit ball. 

The new and exciting perspective on dualities exposes many new research directions, since, for example, one obtains several ways for constructing new order-reversing quasi-involutions from the given ones. In particular, the authors propose a notion of a dual order-reversing quasi-involution and study the dual polarity. From this, a Blaschke-Santaló type inequality for the corresponding Gaussian volume product is obtained. 

Kasia Wyczesany was a Semester Postdoc at ICERM during the Fall 2022 program ‘Harmonic Analysis and Convexity.’ She spent time working on set dualities and their generalizations during her stay in the Institute. 

References 

[1] S. Artstein-Avidan, S. Sadovsky, and K. Wyczesany. A zoo of dualities. arXiv preprint arXiv:2110.11308, 2021.
[2] S. Artstein-Avidan, S. Sadovsky, and K. Wyczesany. A rockafellar-type theorem for non-traditional costs. Advances in Mathematics, 395:108157, 2022.
[3] S. Artstein-Avidan, S. Sadovsky, and K. Wyczesany. Optimal measure transportation with respect to non-traditional costs. Calculus of Variations and Partial Differential Equations, 62(1):35, 2023.