Some remarks on sub-differential calculus
A real-valued function \(f\) on a Banach space \(X\) is Gateaux subdifferentiable at \(x\) if there are linear functional(s) \(p\) in \(X^*\) such that \[ \liminf_{t\rightarrow 0} | t| ^{-1}\{f(x+th) - f(x) - p(th)\} \geq 0 \] for all \(h \in X\setminus\{0\}.\) One writes \(p \in D_G^{-}f(x)\). For superdifferentiability one reverses the inequality and writes \(p \in D_G^{+}f(x)\). The author considers Borel functions \(f\) for which \(D_G^{-}f(x) \cap D_G^{+}f(x) \neq \emptyset\) for all \(x \in X\). He then defines \(\Phi : X \mapsto \mathbb R\) by \[ \Phi (x) := \inf\{\| p\| : p \in D_G^{-}f(x) \cap D_G^{+}f(x)\}. \] The main result is that the Lebesgue measure of the image under \(f\) of any line segment \([x,y]\) in \(X\) is bounded above by the product of \(\| x - y\| \) and a ``lower integral'' over \([0,1]\) of \(\Phi(ty + (1-t)x)\) (as a function of \(t\)). The approach is to use classical results and methods from Theory of the Integral by \textit{S. Saks} [``Theory of the integral'' (1937; Zbl 0017.30004)].
scientific article; zbMATH DE number 1267962