convex envelope

== English == === Noun === convex envelope (plural convex envelopes) (mathematical analysis, of a set) 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 τ ( E , E ′ ) {\displaystyle \tau (E,E')} -complete. 1987, H. G. Eggleston, S. Madan (translators), Nicolas Bourbaki, 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 ∑ i λ i x i {\displaystyle \sum _{i}\lambda _{i}x_{i}} , where ( x i ) {\displaystyle (x_{i})} is any finite family of points in A, the numbers λ i > 0 {\displaystyle \lambda _{i}>0} for all i and ∑ i λ i = 1 {\displaystyle \sum _{i}\lambda _{i}=1} . (mathematics, optimisation theory, of a function on a set) For a given set S ⊆ R n {\displaystyle 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 ==== (optimisation theory): Called the convex envelope of f on S. Notations include conv S ( f ) {\displaystyle {\text{conv}}_{S}(f)} and, assuming S is understood, f ^ {\displaystyle {\hat {f}}} . ==== Synonyms ==== (optimisation theory): lower convex envelope ==== Coordinate terms ==== concave envelope upper concave envelope