Abstract
A subset C of E n is strongly convex if there exists a positive constant k such that for all x and y in C, ( x+ y)/2+δϵ C for all δϵ E n verifying |δ|≦ k| x− y| 2. A function f : E n → E is strongly convex if there exists a constant α>0 such that for all x and y, f(( x+ y)/2)≦ 1 2 ; f( x)+ 1 2 ;( y)−α| x − y| 2. Five characterizations of strongly convex sets are given. The level sets of strongly convex functions are shown to be strongly convex. Moreover it is proved that a function is locally strongly convex if and only if its epigraph is locally strongly convex. Finally the concept of strongly quasi-convex function is given along with a property of its level sets.
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