Spherically Supported Sets with Variable Radii
Abstract
We study closed sets for which, at every boundary point, there exists a containing supporting ball whose prescribed radius may vary from point to point. This extends the notion of spherical support from constant to variable radii. The variable-radius setting presents a new difficulty: unlike the constant-radius case, the intersection of the supporting balls need not be strongly convex with respect to the prescribed radii. We overcome this difficulty by associating with every such set a canonical set and an induced radius function determined by the active supporting balls. For continuous prescribed radii, our main result establishes a geometric dichotomy: the original set either coincides with the canonical set and is strongly convex with respect to the prescribed radii, or it is contained in the boundary of the canonical set, which is strongly convex with respect to the induced radii. An example shows that the induced radii cannot, in general, be replaced by the prescribed ones in the second alternative. We also recover the constant-radius theory and characterize variable-radius strong convexity in terms of the spherical support of the boundary.
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BibTeX
@misc{nour2026spherically,
title = {{Spherically Supported Sets with Variable Radii}},
author = {Chadi Nour},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28789},
eprint = {2610.28789},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28789},
}
RIS
TY - UNPB
TI - Spherically Supported Sets with Variable Radii
AU - Chadi Nour
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28789
UR - https://arxiv.org/abs/2610.28789
AB - We study closed sets for which, at every boundary point, there exists a containing supporting ball whose prescribed radius may vary from point to point. This extends the notion of spherical support from constant to variable radii. The variable-radius setting presents a new difficulty: unlike the constant-radius case, the intersection of the supporting balls need not be strongly convex with respect to the prescribed radii. We overcome this difficulty by associating with every such set a canonical set and an induced radius function determined by the active supporting balls. For continuous prescribed radii, our main result establishes a geometric dichotomy: the original set either coincides with the canonical set and is strongly convex with respect to the prescribed radii, or it is contained in the boundary of the canonical set, which is strongly convex with respect to the induced radii. An example shows that the induced radii cannot, in general, be replaced by the prescribed ones in the second alternative. We also recover the constant-radius theory and characterize variable-radius strong convexity in terms of the spherical support of the boundary.
ER -