Clopen sign approximation in the real Cantor unit ball
= Clopen sign approximation in the real Cantor unit ball
Every real continuous function of supremum norm at most one on the <Cantor set> is a uniform limit of convex combinations of continuous sign functions. Approximate on a finite <Cantor cylinder> partition by values $a_j\in[-1,1]$. The sign vector $s$ has weight $\prod_j(1+s_ja_j)/2$, whose coordinate means are $a_j$. This proves a <closed convex hull> identity without weak compactness.