Semiconvex Function on a Convex Subset of $\mathbb{R}^n$

Let $n\ge1$ be a natural number, let $\mathbb{R}$ be the real numbers, let $C\subseteq\mathbb{R}^n$ be convex, let $f:C\to\mathbb{R}$, and let $\lambda\in\mathbb{R}$ satisfy $0\le\lambda$. Write $\lVert\,\cdot\,\rVert$ for the Euclidean norm on $\mathbb{R}^n$.

We say that $f$ is semiconvex on $C$ with constant $\lambda$ if the function $g:C\to\mathbb{R}$ given by

is convex on $C$.