On the Solution Sets of the Cone-Volume Map

Abstract

For a fixed set U=(u₁,…,u_m) of outer unit normals, we consider the cone-volume map which associates to a vector of support numbers b∈ℝ^m_{> 0} the cone-volume vector γ∈ℝ^m of the polytope P(U,b)={x∈ℝⁿ:U^⊤ x≤ b}. We study its fibers S(U,γ) over prescribed cone-volume vectors γ, with particular emphasis on finiteness and non-uniqueness. We give characterizations of the irreducibility of U in terms of properties of the cone-volume map and its solution sets. In particular, for irreducible U, we show that S(U,γ) is finite for all γ outside a set of measure zero in the cone-volume cone, and we give an explicit upper bound for its cardinality. In the planar case, we prove that S(U,γ) has semialgebraic dimension at most one for arbitrary U. Moreover, for every U∈𝒰(2,m) with m≥ 4, there exists a cone-volume vector γ for which S(U,γ) contains at least two distinct elements. As a consequence, there exist two distinct smooth planar convex bodies with the same cone-volume measure. Finally, for U in general position, we show that the number of solutions is odd for every cone-volume vector outside the exceptional set.

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BibTeX RIS

BibTeX

@misc{baumbach2026solution,
  title = {{On the Solution Sets of the Cone-Volume Map}},
  author = {Tom Baumbach and Zhang Chen and Martin Henk},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.21533},
  eprint = {2610.21533},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.21533},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - On the Solution Sets of the Cone-Volume Map
AU  - Tom Baumbach
AU  - Zhang Chen
AU  - Martin Henk
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.21533
UR  - https://arxiv.org/abs/2610.21533
AB  - For a fixed set U=(u_1,…,u_m) of outer unit normals, we consider the cone-volume map which associates to a vector of support numbers b∈ℝ^m_> 0 the cone-volume vector γ∈ℝ^m of the polytope P(U,b)={x∈ℝ^n:U^⊤ x≤ b}. We study its fibers S(U,γ) over prescribed cone-volume vectors γ, with particular emphasis on finiteness and non-uniqueness. We give characterizations of the irreducibility of U in terms of properties of the cone-volume map and its solution sets. In particular, for irreducible U, we show that S(U,γ) is finite for all γ outside a set of measure zero in the cone-volume cone, and we give an explicit upper bound for its cardinality. In the planar case, we prove that S(U,γ) has semialgebraic dimension at most one for arbitrary U. Moreover, for every U∈𝒰(2,m) with m≥ 4, there exists a cone-volume vector γ for which S(U,γ) contains at least two distinct elements. As a consequence, there exist two distinct smooth planar convex bodies with the same cone-volume measure. Finally, for U in general position, we show that the number of solutions is odd for every cone-volume vector outside the exceptional set.
ER  -

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