Convex envelope

English

Noun

1. Template:Lb Convex hull.
   • 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 $\tau (E,E^{\prime })$-complete.

   • 1987, H. G. Eggleston, S. Madan (translators), Template:W, 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 $\sum _i{\lambda }_i=1$.

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2. Template:Lb For a given set $S\subseteq {ℝ}^n$ and real-valued function f defined on the convex hull conv(S), the highest-valued convex function that underestimates or equals f over S.
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Usage notes

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

Synonyms

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Coordinate terms

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