= Non-vertical supporting hyperplane of a value function
{title2=$r=\phi(b)+\lambda^T(u-b)$}
At a point where $\phi(b)$ is finite, this is an affine plane supporting the <epigraph> from below: $\phi(u)\geq\phi(b)+\lambda^T(u-b)$ for every $u$. A nonzero coefficient of the vertical coordinate permits normalization to the displayed graph form. For a constrained <value function>, the slope is exactly an attained multiplier in the <Strong Lagrangian property>. When the <value function> is differentiable at an interior point of its finite domain, a supporting slope equals its <gradient>: apply the supporting inequality along both positive and negative multiples of each coordinate direction and take limits.
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