The Sup-Convolution of an Upper Semicontinuous Function Attains its Supremum

Let $M$, $\mathbb{R}$, $\mathbb{R}^{M}$ and $\lVert\,\cdot\,\rVert$ be as in Sup-Convolution of a Function on $\mathbb{R}^M$, and let $d$ be the Euclidean distance, a metric on $\mathbb{R}^{M}$.

Let $v:\mathbb{R}^{M}\to\mathbb{R}$ be upper semicontinuous on $\mathbb{R}^{M}$ with respect to $d$, let $C\in\mathbb{R}$ be an upper bound for the set of values of $v$, let $\lambda\in\mathbb{R}$ satisfy $0<\lambda$, let $v^{\lambda}$ be the sup-convolution of $v$ with parameter $\lambda$, and let $\xi\in\mathbb{R}^{M}$. Then the following hold.

1. (Localisation) Every $y\in\mathbb{R}^{M}$ with $v(\xi)\le v(y)-\frac{\lambda}{2}\lVert y-\xi\rVert^{2}$ satisfies

$$\frac{\lambda}{2}\,\lVert y-\xi\rVert^{2}\le C-v(\xi).$$

2. (Attainment) There exists $y\in\mathbb{R}^{M}$ with

$$v^{\lambda}(\xi)=v(y)-\frac{\lambda}{2}\,\lVert y-\xi\rVert^{2},$$

and every such $y$ satisfies the bound of claim 1.