Let f be a real-valued radially lower semicontinuous function defined on a convex subset D of a real vector space. It is shown that f is convex if and only if, for all x, y ∈ D there exists α = α(x, y) ∈ (0, 1) such that f (αx + (1 − α)y) ≤ αf (x) + (1 − α)f (y).
A Characterization of Convex Functions
Leonetti P
2018-01-01
Abstract
Let f be a real-valued radially lower semicontinuous function defined on a convex subset D of a real vector space. It is shown that f is convex if and only if, for all x, y ∈ D there exists α = α(x, y) ∈ (0, 1) such that f (αx + (1 − α)y) ≤ αf (x) + (1 − α)f (y).
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