Basic Properties of the Legendre-Fenchel Conjugate: the Fenchel-Young Inequality, Convexity, Full Domain under Superlinear Growth, and the Quadratic

In the setting of The Real Numbers: Standing Notation and Background, let $n\in\mathbb{N}$, let $C$ be a nonempty subset of Euclidean space $\mathbb{R}^{n}$, let $f:C\to\mathbb{R}$, and let $D(f^{*})$ and $f^{*}:D(f^{*})\to\mathbb{R}$ be the domain and the Legendre-Fenchel conjugate of $f$. Write $p\cdot x$ for the dot product and $\lVert x\rVert$ for the Euclidean norm.

1. (Fenchel-Young inequality)¶ $p\cdot x\le f(x)+f^{*}(p)$ for every $x\in C$ and every $p\in D(f^{*})$.

2. (Convexity)¶ $D(f^{*})$ is a convex subset of $\mathbb{R}^{n}$, and $f^{*}$ is convex on $D(f^{*})$.

3. (Full domain)¶ If for every positive $R\in\mathbb{R}$ there is $c_{R}\in\mathbb{R}$ with $R\lVert x\rVert-c_{R}\le f(x)$ for every $x\in C$, then $D(f^{*})=\mathbb{R}^{n}$.

4. (The quadratic)¶ If $C=\mathbb{R}^{n}$ and $f(x)=\tfrac12\lVert x\rVert^{2}$ for every $x$, then $D(f^{*})=\mathbb{R}^{n}$ and $f^{*}(p)=\tfrac12\lVert p\rVert^{2}$ for every $p\in\mathbb{R}^{n}$.