convex envelope

• 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')-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 ∑ᵢλᵢx_i, where (x_i) is any finite family of points in A, the numbers λᵢ>0 for all i and ∑ᵢλᵢ=1.

Optimisation theory Of a function on a set

• One of the main reasons for the interest in convex envelopes is the fact that the set of global minimum points of f on S is contained in the set of global minimum points of conv_S(f) on S and the two minimum values coincide (see, e.g., [11, 17]). Hence, if the convex envelope were efficiently computable or available in closed form, one could replace the nonconvex problem of minimizing f on S with the convex problem of minimizing f on conv_S(f).2004, Fabio Tardella, “On the existence of polyhedral convex envelopes”, in Christodoulos A. Floudas, Panos M. Pardalos, editors, Frontiers in Global Optimization, Springer (Kluwer Academic), page 564:
• Tawarmalani and Sahinidis (2001) developed the convex envelope and concave envelope for x/y over a unit hypercube, compared it to the convex relaxation proposed by Zamora and Grosmmann (1998a), (1998b), (1999), proposed a semidefinite relaxation of x/y, and suggested convex envelopes for functions of the form f(x)y² and f(x)/y.2004, C. A. Floudas, I. G. Akrotiriankis, S. Caratzoulas, C. A. Meyer, J. Kallrath, “Global Optimization in the 21st Century: Advances and Challenges”, in Ana Paula Barbosa-Póvoa, Henrique Matos, editors, European Symposium on Computer Aided Process Engineering-14: 37th European Symposium of the Working Party, Elsevier, page 25: