PreprintCC BY-NC-ND 4.0arXiv
On determination of convex bodies by their section functions
Abstract
For a convex body K its parallel section function in the direction θ∈ Sn−1 is defined by AK,θ(t) = voln−1 ( K ∩ { x ∈ ℝn | ⟨ x, θ⟩ = t } ) . We study to what extent the body K is determined by partial information about these functions. As applications, we characterize convex bodies with locally separable section functions, establish partial results on the homothety conjecture for bodies of flotation, and answer a question of Barker and Larman concerning the determination of convex bodies from section functions at infinitely many distances from the origin.
Related
- The Petty Conjecture for Convex Bodies of Revolution
- Riemannian convex bodies with a geodesic segment on the boundary
- Continuity and monotonicity of weighted perimeters of convex bodies
- Comment on "The polarization hierarchy for polynomial optimization over convex bodies, with applications to nonnegative matrix rank"
- A Nearly Quadratic Lower Bound for Linear Optimization over Convex Bodies in the Membership Oracle Model
BibTeX
@misc{fish2026determination,
title = {{On determination of convex bodies by their section functions}},
author = {Alexander Fish and Dmitry Ryabogin and Wen Rui Sun and Vladyslav Yaskin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10229},
eprint = {2610.10229},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10229},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - On determination of convex bodies by their section functions
AU - Alexander Fish
AU - Dmitry Ryabogin
AU - Wen Rui Sun
AU - Vladyslav Yaskin
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10229
UR - https://arxiv.org/abs/2610.10229
AB - For a convex body K its parallel section function in the direction θ∈ S^n−1 is defined by A_K,θ(t) = vol_n−1 ( K ∩ { x ∈ ℝ^n | ⟨ x, θ⟩ = t } ) . We study to what extent the body K is determined by partial information about these functions. As applications, we characterize convex bodies with locally separable section functions, establish partial results on the homothety conjecture for bodies of flotation, and answer a question of Barker and Larman concerning the determination of convex bodies from section functions at infinitely many distances from the origin.
ER -