= Solution
For $g\in K$, <convexity> gives $h+t(g-h)\in K$ for $0\leq t\leq1$. Thus $F(t)=\|h-f+t(g-h)\|_p^p$ has a right minimum at $t=0$ and $F'(0)\geq0$. The <derivative of a power of the Lp norm> is
$$
F'(0)=p\,\operatorname{Re}\int_{\mathbb R^n}|h-f|^{p-2}\overline{(h-f)}(g-h)\,dx.
$$
Consequently
$$
\boxed{\operatorname{Re}\int_{\mathbb R^n}|h-f|^{p-2}\overline{(h-f)}(h-g)\,dx\leq0.}
$$
The printed hint has coefficient $2$ where the general coefficient is $p$. This does not affect the desired sign, since $p>0$. The complex conjugation is important for complex-valued functions; for real functions it can be omitted.
Back to article page
