PomeDefinition 21.3.1.103D0D196EAF The Legendre transform, or convex conjugate, of a function $F\colon N_{\mathbb R}\to [-\infty ,+\infty ]$ is the function on $M_{\mathbb R}$ given by
\[ F^*(p)=\sup _{x\in N_{\mathbb R}}(\langle p,x\rangle -F(x)). \]
The conjugate of a function on $M_{\mathbb R}$ is defined in the same way and is a function on $N_{\mathbb R}$; we write $F^{**}=(F^*)^*$. The convex indicator function $\chi _ C$ of a nonempty set $C$, in either space, is $0$ on $C$ and $+\infty $ elsewhere. The support function of a nonempty set $P\subset M_{\mathbb R}$ is
\[ h_ P(x)=\sup _{p\in P}\langle p,x\rangle \qquad (x\in N_{\mathbb R}); \]
it is convex with values in $(-\infty ,+\infty ]$, and it is finite when $P$ is bounded.
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