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Some remarks on sub-differential calculus. (English) Zbl 0923.46044
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 S. Saks [“Theory of the integral” (1937; Zbl 0017.30004)].

##### MSC:
 46G05 Derivatives of functions in infinite-dimensional spaces 49J52 Nonsmooth analysis 46B20 Geometry and structure of normed linear spaces 26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
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