A note on the differentiability of convex functions

Abstract

Every real-valued convex and locally Lipschitzian function <italic>f</italic> defined on a nonempty closed convex set <italic>D</italic> of a Banach space <italic>E</italic> is the local restriction of a convex Lipschitzian function defined on <italic>E</italic>. Moreover, if <italic>E</italic> is separable and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="i n t upper D not-equals normal empty-set"> <mml:semantics> <mml:mrow> <mml:mi>int</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mi>D</mml:mi> <mml:mo>≠<!-- ≠ --></mml:mo> <mml:mi mathvariant="normal">∅<!-- ∅ --></mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {int} D \ne \emptyset</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then, for each Gateaux differentiability point <italic>x</italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis element-of i n t upper D right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mi>int</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mi>D</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">( \in \operatorname {int} D)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <italic>f</italic>, there is a closed convex set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C subset-of i n t upper D"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo>⊂<!-- ⊂ --></mml:mo> <mml:mi>int</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mi>D</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">C \subset \operatorname {int} D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with the nonsupport points set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N left-parenthesis upper C right-parenthesis not-equals normal empty-set"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>C</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>≠<!-- ≠ --></mml:mo> <mml:mi mathvariant="normal">∅<!-- ∅ --></mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">N(C) \ne \emptyset</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x element-of upper N left-parenthesis upper C right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>C</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">x \in N(C)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Subscript upper C"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>C</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{f_C}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (the restriction of <italic>f</italic> on <italic>C</italic>) is Fréchet differentiable at <italic>x</italic>.