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I'm reading Section 2.1 of Chapter 2 in Villani's textbook Topics in Optimal Transportation. enter image description here

Let $X := \mathbb R^d$, $\mu$ a Borel probability measure on $X$, and $\varphi : X \to \mathbb R \cup \{+\infty\}$ convex. A subset $A$ of $X$ is called a small set if and only if the Hausdorff dimension of $A$ is at most $d-1$. Let $D := \{x \in X \mid \varphi (x) < +\infty\}$ be the domain of $f$. Then $D$ is convex.

Theorem: Let $E := \{x \in \operatorname{int} (D) \mid \varphi \text{ is not Fréchet differentiable at }x\}$. If $\varphi \in L_1 (\mu)$, then $E$ is measurable and a small set.

It follows from $\varphi \in L_1 (\mu)$ that $D$ is measurable and $\mu (D) =1$. Let $K := \operatorname{int} (D)$ and $$ F := \{x \in K \mid \varphi \text{ is Fréchet differentiable at }x\}. $$

Clearly, $E$ is measurable if and only if $F$ is measurable. In case $d=1$, we can prove that $F$ is measurable and $E$ is countable. This directly implies the theorem.

Could you elaborate on or provide a reference of how to prove the theorem in case $d > 1$?

Akira
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1 Answers1

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I have found $2$ below references.

  • Luiro, Hannes. "On the differentiability of directionally differentiable functions and applications." arXiv preprint arXiv:1208.3971 (2012).

It is not difficult to see (see e.g. [ACP]) that the set of non-differentiability of a directionally differentiable Lipschitz-function $f$ is $\sigma$-porous, thus it can be included in a countable union of sets $E_{i}$ such that for any $x \in E_{i}$ there exists $0<\delta_{i}(x)<\frac{1}{2}$ so that for any $0<r<1$ there exists a ball $B(y, \delta r) \subset$ $B(x, r) \backslash E_{i}$. More careful analysis shows that actually $\delta(x)$ can be chosen independently on $x$ or $i$ and also arbitrarily close to $\frac{1}{2}$. This argument also implies the obviously best Hausdorff dimension estimate, $n-1$, for the set of non-differentiability.

  • Mirzaie, Reza. (2017). Hausdorff dimension of the nondifferentiability set of a convex function. Filomat. 31. 5827-5831. 10.2298/FIL1718827M.

The set of nondifferentiability points of a directionally differentiable Lipschitz-function $f$ defined on $R^{n}$ is $\sigma$-porous (see [1]). Thus, it can be included in a countable union of sets $E_{i}$ with the property that for all $x \in E_{i}$ and all $0<r<1$, there exists $0<\delta_{i}(x)<\frac{1}{2}$ such that a ball $B(y, \delta r)$ is included in $B(x, r)-E_{i}$. This argument also implies the best Hausdorff dimension estimate, $n-1$, for the set of nondifferentiability.

Akira
  • 17,367