convex envelope

Noun

 * 1)  Convex hull.
 * 2) * 1965 [Holt Rinehart & Winston], Robert E. Edwards, Functional Analysis: Theory and Applications, Dover, 1995, Unabridged Corrected Edition, page 561,
 * In E the closed convex envelope of a compact (resp. weakly compact) set is $$\tau(E, E')$$-complete.
 * 1) * 1987, H. G. Eggleston, S. Madan (translators),, Topological Vector Spaces: Chapters 1–5, [1981, N. Bourbaki, Espaces Vectoriels Topologiques], Springer, page IR-10,
 * Corollary 1. — The convex envelope of a subset A of E is identical with the set of linear combinations $$\sum_i\lambda_i x_i$$, where $$(x_i)$$ is any finite family of points in A, the numbers $$\lambda_i > 0$$ for all i and $$\sum_i \lambda_i = 1$$.
 * 1)  For a given set $$S\subseteq\mathbb{R}^n$$ and real-valued function f defined on the convex hull conv(S), the highest-valued convex function that underestimates or equals f over S.
 * 1)  For a given set $$S\subseteq\mathbb{R}^n$$ and real-valued function f defined on the convex hull conv(S), the highest-valued convex function that underestimates or equals f over S.

Usage notes
Called the convex envelope of f on S. Notations include $$\text{conv}_S(f)$$ and, assuming S is understood, $$\hat f$$.