Blaschke--Santaló inequalities with respect to costs

@misc{kolesnikov2026blaschkesantaló,
  title = {{Blaschke--Santaló inequalities with respect to costs}},
  author = {Alexander V. Kolesnikov and Galyna V. Livshyts and Liran Rotem},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.19085},
  eprint = {2610.19085},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.19085},
}

TY  - UNPB
TI  - Blaschke--Santaló inequalities with respect to costs
AU  - Alexander V. Kolesnikov
AU  - Galyna V. Livshyts
AU  - Liran Rotem
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.19085
UR  - https://arxiv.org/abs/2610.19085
AB  - 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.
ER  -