Binary digit-sum inequality
= Binary digit-sum inequality
{title2=$F(a+b)\geq F(a)+F(b)+\min(a,b)$}
For $F(a)=\sum_{j=0}^{a-1}s_2(j)$ and nonnegative integers $a,b$, the displayed inequality follows by induction from $F(2r)=2F(r)+r$ and $F(2r+1)=F(r)+F(r+1)+r$. Splitting a vertex set of a <hypercube graph> into its two coordinate sections then proves the <edge-isoperimetric theorem for binary initial segments>.