Sentence:  One of the main reasons for the interest in <WSD>convex envelopes</WSD> 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<sub>S</sub>"("f") on "S" and the two minimum values coincide ("see", e.g., [11, 17]). Hence, if the <WSD>convex envelope</WSD> 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<sub>S</sub>"("f"). 
Meaning of the word marked with <WSD> tags:  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".