Blaschke--Santaló inequalities with respect to costs

Abstract

We study functional Blaschke--Santaló inequalities in which the inner product is replaced by a cost. An inequality on each orthant, combined with Steiner symmetrization, gives a sufficient condition for a cost inequality to hold for even convex functions. We apply this principle to a multiplicative cost defined on all orthants. Its second variation gives a weighted improvement of the Brascamp--Lieb inequality in the directions for which differentiation is justified. We also prove an $L^p$ version of the Fradelizi--Meyer theorem from a weighted volume-product inequality and derive the corresponding radial cost and Brascamp--Lieb inequalities. Explicit examples include anisotropic power and Cauchy densities, with dilation equality cases and a sharp obstruction below the coordinate exponent two. For the spherical logarithmic cost we compute an exact entropy--logdet deficit decomposition and the second-order deficit transverse to ellipsoids. We also explain the transport-entropy formulation and distinguish the restricted spectral question from the scalar $(B)$ property.