PreprintCC BY 4.0arXiv
An approximate inner ball condition characterizing C1,α convex bodies
Abstract
The main result of this paper uses an approximate inner ball condition to characterize C1,α convex bodies for 0<α≤ 1. In particular, we show that a convex body K is of class C1,α if and only if there exist constants C,ε0>0 such that for every 0<ε≤ε0 and for every x∈∂ K there exists hε(x) satisfying B(hε(x), Cε/1−α1+α)⊂ K and dist(x,B(hε(x), Cε/1−α1+α))≤ε. We then give a new proof of the regularity of the convex envelope by showing that conv(f)∈ C1,αloc(ℝn) whenever f∈ C1,αloc(ℝn) and f satisfies the coercivity condition f(x)→∞ as |x|→∞. Finally, we use the approximate inner ball characterization of C1,α convex bodies to prove that the Minkowski sum of a C1,α convex body and a nonempty closed convex set is a C1,α convex domain.
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BibTeX
@misc{cappello2026approximate,
title = {{An approximate inner ball condition characterizing $C^1,α$ convex bodies}},
author = {Anthony R. Cappello},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36238},
eprint = {2610.36238},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36238},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An approximate inner ball condition characterizing C^1,α convex bodies
AU - Anthony R. Cappello
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.36238
UR - https://arxiv.org/abs/2610.36238
AB - The main result of this paper uses an approximate inner ball condition to characterize C^1,α convex bodies for 0<α≤ 1. In particular, we show that a convex body K is of class C^1,α if and only if there exist constants C,ε_0>0 such that for every 0<ε≤ε_0 and for every x∈∂ K there exists h^ε(x) satisfying B(h^ε(x), Cε^/1−α1+α)⊂ K and dist(x,B(h^ε(x), Cε^/1−α1+α))≤ε. We then give a new proof of the regularity of the convex envelope by showing that conv(f)∈ C^1,α_loc(ℝ^n) whenever f∈ C^1,α_loc(ℝ^n) and f satisfies the coercivity condition f(x)→∞ as |x|→∞. Finally, we use the approximate inner ball characterization of C^1,α convex bodies to prove that the Minkowski sum of a C^1,α convex body and a nonempty closed convex set is a C^1,α convex domain.
ER -