Jensen's Lemma: the Contact Set of a Semiconvex Function at a Strict Maximum has Positive Measure

Let $n$ be a natural number with $1\le n$ and let $\mathbb{R}$ be the real numbers with the order $\le$ of their ordered field structure; $2$ denotes $1+1$, which satisfies $0<2$ and so has a multiplicative inverse by claim 8 of Elementary Order Arithmetic in an Ordered Field, and $\tfrac{a}{2}$ denotes the product of $a$ with that inverse. Regard Euclidean space $\mathbb{R}^{n}$ as a real vector space, with the sum of points, the scalar multiple, and the difference $y-x$ and dot product $p\cdot x$ of points; write $\lVert\,\cdot\,\rVert$ for the Euclidean norm and $d_{E}$ for the Euclidean distance, a metric on $\mathbb{R}^{n}$ with $d_{E}(x,y)=\lVert x-y\rVert$ by claim 2 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square. Open balls $B_{d_{E}}$ are those of Open Ball in a Metric Space and closed balls $\bar{B}_{d_{E}}$ those of Closed Ball in a Metric Space; $0_{\mathbb{R}^{n}}$ is the origin. Compactness refers to the topology of the open sets of $(\mathbb{R}^{n},d_{E})$, a topology by The Open Subsets of a Metric Space Form a Topology. Let $\mathcal{B}(\mathbb{R}^{n})$ be the Borel $\sigma$-algebra and $\lambda_{n}$ Lebesgue measure on it. Powers with natural exponent are those of Natural Number Power of an Element of a Field, and $\sigma_{n}$ is the positive real number with $\sigma_{n}^{2}=n$ introduced in Uniform Grids on a Half-Open Box and Grid Hulls of a Compact Set in $\mathbb{R}^n$.

Let $U\subseteq\mathbb{R}^{n}$ be convex, let $\lambda\in\mathbb{R}$ satisfy $0<\lambda$, and let $\varphi:U\to\mathbb{R}$ be semiconvex on $U$ with constant $\lambda$. Let $\hat{x}\in\mathbb{R}^{n}$ and $r\in\mathbb{R}$ satisfy $0<r$ and $\bar{B}_{d_{E}}(\hat{x},r)\subseteq U$; write $\bar{B}=\bar{B}_{d_{E}}(\hat{x},r)$, and suppose that $\hat{x}$ is a strict maximum point of $\varphi$ on $\bar{B}$, in the sense that

$$\varphi(x)<\varphi(\hat{x})\qquad\text{for every }x\in\bar{B}\text{ with }x\ne\hat{x}.$$

Suppose further that $U$ is open in $(\mathbb{R}^{n},d_{E})$. For $\delta\in\mathbb{R}$ with $0<\delta$ put

$$K_{\delta}=\Bigl\{x\in\bar{B}\ :\ \text{there is }p\in\mathbb{R}^{n}\text{ with }\lVert p\rVert\le\delta\text{ and }\varphi(y)+p\cdot y\le\varphi(x)+p\cdot x\text{ for every }y\in\bar{B}\Bigr\},$$

which is the contact set $K_{\delta}$ of Maximisers of Linearly Perturbed Continuous Functions on a Closed Ball: Existence, Localisation, and Compactness of the Contact Set formed from the restriction of $\varphi$ to $\bar{B}$.

Then there is $\delta_{0}\in\mathbb{R}$ with $0<\delta_{0}$ such that the following hold.

1. (Compact contact set in the half ball) (i) For every real $\delta$ with $0<\delta\le\delta_{0}$, the set $K_{\delta}$ is nonempty, is compact in $\mathbb{R}^{n}$, belongs to $\mathcal{B}(\mathbb{R}^{n})$, and satisfies $K_{\delta}\subseteq\bar{B}_{d_{E}}\bigl(\hat{x},\tfrac{r}{2}\bigr)$.

2. (Positive measure) (ii) For every real $\delta$ with $0<\delta\le\delta_{0}$,

$$\lambda_{n}\bigl(\bar{B}_{d_{E}}(0_{\mathbb{R}^{n}},\delta)\bigr)\ \le\ \bigl(4\,\sigma_{n}\,\lambda\bigr)^{n}\,\lambda_{n}(K_{\delta}),$$

and consequently $0<\lambda_{n}(K_{\delta})$.

3. (The contact set lies in no null set) (iii) For every real $\delta$ with $0<\delta\le\delta_{0}$ and every $Z\in\mathcal{B}(\mathbb{R}^{n})$ with $\lambda_{n}(Z)=0$, there is $x\in K_{\delta}$ with $x\notin Z$.

4. (Concentration at the maximum point) (iv) For every $\rho\in\mathbb{R}$ with $0<\rho$ there is $\delta_{1}\in\mathbb{R}$ with $0<\delta_{1}\le\delta_{0}$ such that

$$K_{\delta}\subseteq B_{d_{E}}(\hat{x},\rho)\qquad\text{for every real }\delta\text{ with }0<\delta\le\delta_{1}.$$