Alexandrov--Fenchel inequalities for weighted curvature integrals in space forms

@misc{xu2026alexandrovfenchel,
  title = {{Alexandrov--Fenchel inequalities for weighted curvature integrals in space forms}},
  author = {Botong Xu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.40087},
  eprint = {2610.40087},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.40087},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Alexandrov--Fenchel inequalities for weighted curvature integrals in space forms
AU  - Botong Xu
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.40087
UR  - https://arxiv.org/abs/2610.40087
AB  - We establish two types of Alexandrov--Fenchel inequalities for the weighted curvature integrals of static convex domains in space forms. The first type consists of a complete family of inequalities involving three weighted curvature integrals, with the second Minkowski type inequality of Xia [Calc. Var. Partial Differential Equations 55 (2016), Art. 96] and the weighted almost Schur lemma of Li and Xia [J. Differential Geom. 113 (2019), pp. 493--518] as special cases. The second type consists of a complete family of inequalities involving two weighted curvature integrals, confirming a conjectured geometric inequality by Hu and Li [Trans. Amer. Math. Soc. 375 (2022), pp. 5587--5615]. The key novelty lies in observing that, via the orthogonal projection map, the weighted curvature integrals for domains in space forms admit a natural interpretation in terms of mixed volumes of Euclidean convex bodies, thus reducing the problem to the classical Alexandrov--Fenchel inequalities in convex geometry.
ER  -