Maximisers of Linearly Perturbed Continuous Functions on a Closed Ball: Existence, Localisation, and Compactness of the Contact Set

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 and the absolute value $|\cdot|$. Regard Euclidean space $\mathbb{R}^{n}$ as a real vector space, with the sum of points, the scalar multiple, and the difference $x-y$ 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; 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 $\hat{x}\in\mathbb{R}^{n}$, let $r\in\mathbb{R}$ with $0<r$, and write $\bar{B}=\bar{B}_{d_{E}}(\hat{x},r)$. Let $\varphi:\bar{B}\to\mathbb{R}$ satisfy the following two hypotheses.

(H1) (Continuity) (i) For every $x\in\bar{B}$ and every $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$ there is $\delta\in\mathbb{R}$ with $0<\delta$ such that every $y\in\bar{B}$ with $\lVert y-x\rVert<\delta$ satisfies $|\varphi(y)-\varphi(x)|<\varepsilon$.

(H2) (Strict maximum at the centre) (ii) $\varphi(x)<\varphi(\hat{x})$ for every $x\in\bar{B}$ with $x\ne\hat{x}$.

For $p\in\mathbb{R}^{n}$ put

$$M(p)=\bigl\{x\in\bar{B}\ :\ \varphi(y)+p\cdot y\le\varphi(x)+p\cdot x\ \text{ for every }y\in\bar{B}\bigr\},$$

and for $\delta\in\mathbb{R}$ with $0<\delta$ put

$$K_{\delta}=\bigl\{x\in\bar{B}\ :\ x\in M(p)\ \text{ for some }p\in\mathbb{R}^{n}\text{ with }\lVert p\rVert\le\delta\bigr\}.$$

Then the following hold.

1. (Existence of maximisers) (iii) $M(p)\ne\varnothing$ for every $p\in\mathbb{R}^{n}$; consequently $K_{\delta}\ne\varnothing$ for every real $\delta>0$.

2. (Localisation) (iv) For every $\rho\in\mathbb{R}$ with $0<\rho$ there is $\delta_{\rho}\in\mathbb{R}$ with $0<\delta_{\rho}$ such that

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

3. (Compactness of the contact set) (v) $K_{\delta}$ is compact in $\mathbb{R}^{n}$ for every real $\delta>0$.