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PreprintCC BY 4.0HAL

DISCRETE MIXED VOLUMES AND ALEXANDROV–FENCHEL-TYPE INEQUALITIES

  • Joel Hakavuori
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Abstract

Motivated by the parallels between volume and lattice point counting, we study the mixed version of Ehrhart theory, computing the number of lattice points in Minkowski sums of lattice polytopes. This quantity is a polynomial in the scalars, and the coefficients in the binomial basis are given by the discrete mixed volumes, which generalize mixed volumes in the case of lattice polytopes. We prove a general asymptotic Alexandrov-Fenchel-type inequality for the discrete mixed volumes of lattice polytopes, and an exact inequality for coordinate simplices. To prove the exact inequality, we use the exceptional Hirzebruch-Riemann-Roch theorem of Berget-Eur-Spink-Tseng on the permutohedral variety to express the discrete mixed volume as a classical mixed volume. Additionally, we give a matroidal interpretation of this result, showing that these quantities are exactly the coefficients of the characteristic polynomial of the associated cotransversal matroid, recovering their log-concavity.

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Cite

BibTeXRIS
BibTeX
@misc{hakavuori2026discrete,
  title = {{DISCRETE MIXED VOLUMES AND ALEXANDROV–FENCHEL-TYPE INEQUALITIES}},
  author = {Joel Hakavuori},
  howpublished = {HAL},
  year = {2026},
  month = oct,
  url = {https://hal.science/hal-05774751v1},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - DISCRETE MIXED VOLUMES AND ALEXANDROV–FENCHEL-TYPE INEQUALITIES
AU  - Joel Hakavuori
PY  - 2026
DA  - 2026/10/01
PB  - HAL
UR  - https://hal.science/hal-05774751v1
AB  - Motivated by the parallels between volume and lattice point counting, we study the mixed version of Ehrhart theory, computing the number of lattice points in Minkowski sums of lattice polytopes. This quantity is a polynomial in the scalars, and the coefficients in the binomial basis are given by the discrete mixed volumes, which generalize mixed volumes in the case of lattice polytopes. We prove a general asymptotic Alexandrov-Fenchel-type inequality for the discrete mixed volumes of lattice polytopes, and an exact inequality for coordinate simplices. To prove the exact inequality, we use the exceptional Hirzebruch-Riemann-Roch theorem of Berget-Eur-Spink-Tseng on the permutohedral variety to express the discrete mixed volume as a classical mixed volume. Additionally, we give a matroidal interpretation of this result, showing that these quantities are exactly the coefficients of the characteristic polynomial of the associated cotransversal matroid, recovering their log-concavity.
ER  -