Abstract

In a recent article [Functions on a convex set which are both ω\omega-semiconvex and ω\omega-semiconcave I, J. Convex Analysis 29 (2022) 837–856] we proved with L. Zajíček that if G⊂RnG\subset\mathbb{R}^n is an unbounded open convex set that does not contain a translation of a convex cone with non-empty interior, then there exist f:G→Rf:G\to\mathbb{R} and a concave modulus ω\omega such that lim⁡t→∞ω(t)=∞\lim_{t\to\infty}\omega(t)=\infty, ff is both semiconvex and semiconcave with modulus ω\omega and f∉C1,ω(G)f\notin C^{1,\omega}(G). Here we improve the previous result as follows: If GG is as above and ω(t)=tα\omega(t)=t^{\alpha} for some α∈(0,1)\alpha\in(0,1), then there exists f:G→Rf:G\to\mathbb{R} that is both semiconvex and semiconcave with modulus ω\omega and f∉C1,α(G)f\notin C^{1,\alpha}(G). This result has immediate consequences concerning a first-order quantitative converse Taylor theorem and the problem whether f∈C1,α(G)f\in C^{1,\alpha}(G) whenever ff is smooth in a corresponding sense on all lines.

Author information

Contact details are reproduced from the original publication and may be historical.

Václav Kryštof

Charles University, Faculty of Mathematics and Physics, Praha, Karlín, Czech Republic

vaaclav.krystof@gmail.com

Keywords

  • Semiconvex function with general modulus
  • semiconcave function with general modulus
  • C1,α function
  • C1,ω function
  • unbounded open convex set

Mathematics Subject Classification

26B25

References

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Suggested citation

V. Kryštof. “Functions on a Convex Set which are Both ω-Semiconvex and ω-Semiconcave II.” Journal of Convex Analysis 32 (2025), No. 2, 447–466. https://doi.org/10.68381/jca32022

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Published by Heldermann Verlag, 2025. Rights now held by Banach Press.