Let $f\in C([0,1])$ and $K\subset C([0,1])$ be the set of constant functions on $[0,1]$. Let $\|u\|=\sup\{|u(x)|:\ x\in [0,1]\}$. Define $F:C([0,1])\rightarrow \mathbb{R}$ by $F(g)=\|f-g\|$
Consider the problem of minimize $F$ in $K$ and let $c\in K$ be the minimum.
1 - Is it possible to characterize $c$ in terms of $f$.
2 - Is it possible to characterize $F(c)$.