The Spherical Hadwiger Theorem
Abstract
We prove the spherical Hadwiger theorem in every dimension: for n≥1, every continuous SO(n+1)-invariant valuation on all closed spherical convex sets in the unit sphere 𝕊n is a unique linear combination of the spherical intrinsic volumes V0,…,Vn. The same classification holds on sets contained in an open hemisphere, with unique continuous invariant extension to the full domain. We first classify simple invariant valuations by an independent induction on dimension. Averaging a signed cone identity gives the spherical volume term; the remaining terms are signed coning transforms that vanish by the lower-dimensional simple classification. A second induction, together with extension to non-proper sets, gives the general classification. The cone--sphere correspondence yields, for d≥2, the corresponding classification on all closed convex cones in ℝd, without normalization at the zero cone. In both settings, SO-invariance implies O-invariance.
Related
- Monotone Hadwiger Theorems on Spherical and Hyperbolic Convex Sets
- Hadwiger's Theorem on hyperbolic space
- Hadwiger's classification theorem on the sphere via signed orthoscheme decompositions
- Tweaking the constant in the Linear Hadwiger Theorem
- A Local Fourier-Transform Proof of Offen's Spherical Transfer Theorem
BibTeX
@misc{wang2026spherical,
title = {{The Spherical Hadwiger Theorem}},
author = {Suijie Wang and Shengguo Wu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.27305},
eprint = {2608.27305},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27305},
}
RIS
TY - UNPB
TI - The Spherical Hadwiger Theorem
AU - Suijie Wang
AU - Shengguo Wu
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2608.27305
UR - https://arxiv.org/abs/2608.27305
AB - We prove the spherical Hadwiger theorem in every dimension: for n≥1, every continuous SO(n+1)-invariant valuation on all closed spherical convex sets in the unit sphere 𝕊^n is a unique linear combination of the spherical intrinsic volumes V_0,…,V_n. The same classification holds on sets contained in an open hemisphere, with unique continuous invariant extension to the full domain. We first classify simple invariant valuations by an independent induction on dimension. Averaging a signed cone identity gives the spherical volume term; the remaining terms are signed coning transforms that vanish by the lower-dimensional simple classification. A second induction, together with extension to non-proper sets, gives the general classification. The cone--sphere correspondence yields, for d≥2, the corresponding classification on all closed convex cones in ℝ^d, without normalization at the zero cone. In both settings, SO-invariance implies O-invariance.
ER -