Functions on a Convex Set which are both ω-Semiconvex and ω-Semiconcave

Abstract

<jats:p> Let <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">G \subset \mathbb{R}^n</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> be an open convex set which is either bounded or contains a translation of a convex cone with nonempty interior. It is known that, for every modulus <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\omega</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ω</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , every function on <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">G</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> which is both semiconvex and semiconcave with modulus <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\omega</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ω</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> is (globally) <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">C^{1,\omega}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo separator="true">,</mml:mo> <mml:mi>ω</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -smooth. We show that this result is optimal in the sense that the assumption on <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">G</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> cannot be relaxed. We also present direct short proofs of the above mentioned result and of some its quantitative versions. Our results have immediate consequences concerning (i) a first-order quantitative converse Taylor theorem and (ii) the problem whether <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">f\in C^{1,\omega}(G)</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo separator="true">,</mml:mo> <mml:mi>ω</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> whenever <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">f</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> is continuous and smooth in a corresponding sense on all lines. We hope that these consequences are of an independent interest. </jats:p>