PreprintarXiv licencearXiv
Shades and Convexity of Hypersurfaces
Abstract
Let f: Mn→Rn+1, n≥2, be a smooth immersion of a closed oriented manifold, with Gauss map ν: M→Sn. For each unit vector u, the shade Su⊂ M is the set of points where ν makes an acute angle with u. We show that f is a convex embedding if and only if every shade is contractible. More generally, if every shade is k-connected, then f is a convex embedding provided that n≤ 2k+1, or n=2k+2 and Hk+1(M;Z)=0. On the other hand, for every n≥ 2k+3 there is a nonconvex embedding f:Sn→Rn+1 all of whose shades are k-connected. These results give sharp answers to a question of H. Wente in all dimensions.
Related
- A convexity theorem for proper Dupin hypersurfaces in spheres
- Polynomial convexity and non-polynomial convexity of totally real discs in ℂ2
- The Porous Medium Equation and Convexity
- Convexity of a ratio function involving polygamma functions and monotonicity of the sectional curvature of the Fisher--Rao manifold
- A Deterministic Game-Theoretic Approach to Convexity-Preserving Property and Finite-Time Extinction of Level Sets for Geometric Equations
BibTeX
@misc{ghomi2026shades,
title = {{Shades and Convexity of Hypersurfaces}},
author = {Mohammad Ghomi},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08415},
eprint = {2611.08415},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08415},
}
RIS
TY - UNPB
TI - Shades and Convexity of Hypersurfaces
AU - Mohammad Ghomi
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08415
UR - https://arxiv.org/abs/2611.08415
AB - Let f: M^n→R^n+1, n≥2, be a smooth immersion of a closed oriented manifold, with Gauss map ν: M→S^n. For each unit vector u, the shade S_u⊂ M is the set of points where ν makes an acute angle with u. We show that f is a convex embedding if and only if every shade is contractible. More generally, if every shade is k-connected, then f is a convex embedding provided that n≤ 2k+1, or n=2k+2 and H_k+1(M;Z)=0. On the other hand, for every n≥ 2k+3 there is a nonconvex embedding f:S^n→R^n+1 all of whose shades are k-connected. These results give sharp answers to a question of H. Wente in all dimensions.
ER -