= Strengthened Weierstrass condition
{title2=$E(x,y,p,q)\geq0$}
For a $C^1$ curve $u$, some $\delta>0$ must make the <Weierstrass excess function> nonnegative for every $x\in[a,b]$, every trial slope $q\in\mathbb R$, and all $y,p$ satisfying $|y-u(x)|<\delta$, $|p-u'(x)|<\delta$. Thus the supporting-line property persists in a tube of nearby heights and reference slopes. The term “strengthened” here refers to this neighborhood requirement; strict excess for $q\ne p$ is a further condition.
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