## Convex conjugate

I wasn’t aware that convex conjugate is just Legendre transformation. That is, $latex f^{*}(p)=\sup _{\tilde {x}}\{\langle p,{\tilde {x}}\rangle -f({\tilde {x}})\}\geq \langle p,x\rangle -f(x)$   Note that it is also known as the Legendre-Fenchel transformation. Btw, we have the Fenchel inequality $latex \langle p,x\rangle \le f(x)+f^*(p)$ directly from the definition since \$latex f^{*}(p)=\sup _{\tilde {x}}\{\langle p,{\tilde…