The Legendre-Fenchel Conjugate of a Real Function on a Subset of Euclidean Space

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 $p\cdot x$ be the dot product of $p,x\in\mathbb{R}^{n}$.

1. (Domain)¶ The domain of the conjugate of $f$ is the set $D(f^{*})$ of those $p\in\mathbb{R}^{n}$ for which the map $C\to\mathbb{R}$, $x\mapsto p\cdot x-f(x)$, is bounded above.

2. (Conjugate)¶ The Legendre-Fenchel conjugate of $f$ is the map $f^{*}:D(f^{*})\to\mathbb{R}$,

the least upper bound of a nonempty set of real numbers bounded above (nonempty as $C$ is), which exists by The Real Numbers: Standing Notation and Background §bounds.