Domination, Monotonicity and Semiconvexity of the Sup-Convolution

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 $x\cdot y$ denote the dot product of points of $\mathbb{R}^{M}$. The set $\mathbb{R}^{M}$ is a convex subset of itself, directly from that definition.

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

1. (Domination and upper bound) $v(\xi)\le v^{\lambda}(\xi)$ and $v^{\lambda}(\xi)\le C$.

2. (Monotonicity in the parameter) If $\lambda'\in\mathbb{R}$ satisfies $\lambda\le\lambda'$ and $v^{\lambda'}$ is the sup-convolution of $v$ with parameter $\lambda'$, then $v^{\lambda'}(\xi)\le v^{\lambda}(\xi)$.

3. (Semiconvexity) $v^{\lambda}$ is semiconvex on $\mathbb{R}^{M}$ with constant $\lambda$.