The Almost-Everywhere Gradient of a Lipschitz Semiconvex or Semiconcave Function Is Tangent at Every Absolutely Continuous Measure

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, for $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ let $T_{\mu}$ be the tangent space at $\mu$. Let $L,K\in\mathbb{R}$ be nonnegative and let $w:\mathbb{R}^{d}\to\mathbb{R}$ be bounded and Lipschitz with constant $L$ from $\mathbb{R}^{d}$ with the Euclidean distance to $\mathbb{R}$ with the metric of The Absolute Value Metric on the Real Line. Let $E_{w}$ be the set of those $x\in\mathbb{R}^{d}$ at which $w$, regarded as a map into $\mathbb{R}^{1}$, is differentiable; for $x\in E_{w}$ all partial derivatives of $w$ exist at $x$ by claim 1 of A Derivative Matrix is the Jacobian Matrix, and is Unique, and we let $Dw(x)\in\mathbb{R}^{d}$ be the gradient of $w$ at $x$; and let $\nabla w:\mathbb{R}^{d}\to\mathbb{R}^{d}$ be the map equal to $Dw(x)$ at $x\in E_{w}$ and to $0_{\mathbb{R}^{d}}$ at every $x\in\mathbb{R}^{d}\setminus E_{w}$. Semiconvexity is on $\mathbb{R}^{d}$, a convex set directly from that definition, and $-w$ is the function with value $-w(x)$ at $x$.

1. (Measurability and bound)¶ $E_{w}\in\mathcal{B}(\mathbb{R}^{d})$, the map $\nabla w$ is Borel, and $\lVert\nabla w(x)\rVert\le L$ for every $x\in\mathbb{R}^{d}$; so for every $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ the class of $\nabla w$ belongs to $L^{2}(\mu;\mathbb{R}^{d})$, and is again written $\nabla w$. Moreover $\mu(\mathbb{R}^{d}\setminus E_{w})=0$ for every $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ that is absolutely continuous.

2. (Semiconvex functions)¶ Suppose that $w$ is semiconvex with constant $K$. Then $\nabla w\in T_{\mu}$ for every absolutely continuous $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$.

3. (Semiconcave functions)¶ Suppose that $-w$ is semiconvex with constant $K$. Then $\nabla w\in T_{\mu}$ for every absolutely continuous $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$.