PreprintCC BY 4.0arXiv
A convexity criterion via the De Giorgi slope
Abstract
Let X be a Banach space and f∈𝒞1(X) be bounded from below. We show that if for some m≥ 1, the function x↦ |∇ f(x)|m is convex, then f is convex. We also establish a more general version of this result: if f is continuous and bounded from below, then it is convex, provided x↦ sf(x)m is convex for some m≥ 1, where sf denotes the (De Giorgi) metric slope of f.
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BibTeX
@misc{boulmezaoud2026convexity,
title = {{A convexity criterion via the De Giorgi slope}},
author = {Tahar Z. Boulmezaoud and Aris Daniilidis and Trí Minh Lê},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.01154},
eprint = {2606.01154},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.01154},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A convexity criterion via the De Giorgi slope
AU - Tahar Z. Boulmezaoud
AU - Aris Daniilidis
AU - Trí Minh Lê
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2606.01154
UR - https://arxiv.org/abs/2606.01154
AB - Let X be a Banach space and f∈𝒞^1(X) be bounded from below. We show that if for some m≥ 1, the function x↦ |∇ f(x)|^m is convex, then f is convex. We also establish a more general version of this result: if f is continuous and bounded from below, then it is convex, provided x↦ s_f(x)^m is convex for some m≥ 1, where s_f denotes the (De Giorgi) metric slope of f.
ER -