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Centro-sectional measures for log-concave functions

  • Károly J. Böröczky
  • Jinrong Hu
  • Jiaqian Liu
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Abstract

We introduce centro-sectional measures with parameters q,m for log-concave functions on Rn, defined in terms of the q-th moments of their Radon transforms with respect to the Haar measure on m-dimensional subspaces, where m=1,...,n-1, and establish the corresponding variational formulas. Our measures generalize the notion of dual curvature measure if q=1, and are related to the Sine transform if q=2. In line with the coarea formula for log-concave functions as BV functions, the variational formulas give rise to the Euclidean centro-sectional measures and the spherical centro-sectional measures. In the symmetric setting, we solve the associated even functional centro-sectional Minkowski problem, which asks which pairs of measures can arise as the centro-sectional measures of an even log-concave function.

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Cite

BibTeXRIS
BibTeX
@misc{boroczky2026centrosectional,
  title = {{Centro-sectional measures for log-concave functions}},
  author = {Károly J. Böröczky and Jinrong Hu and Jiaqian Liu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.18553},
  eprint = {2609.18553},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.18553},
}
RIS
TY  - UNPB
TI  - Centro-sectional measures for log-concave functions
AU  - Károly J. Böröczky
AU  - Jinrong Hu
AU  - Jiaqian Liu
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.18553
UR  - https://arxiv.org/abs/2609.18553
AB  - We introduce centro-sectional measures with parameters q,m for log-concave functions on Rn, defined in terms of the q-th moments of their Radon transforms with respect to the Haar measure on m-dimensional subspaces, where m=1,...,n-1, and establish the corresponding variational formulas. Our measures generalize the notion of dual curvature measure if q=1, and are related to the Sine transform if q=2. In line with the coarea formula for log-concave functions as BV functions, the variational formulas give rise to the Euclidean centro-sectional measures and the spherical centro-sectional measures. In the symmetric setting, we solve the associated even functional centro-sectional Minkowski problem, which asks which pairs of measures can arise as the centro-sectional measures of an even log-concave function.
ER  -