A function f : C → R on a convex set C is convex iff f(λx + (1−λ)y) ≤ λf(x) + (1−λ)f(y) for all x, y ∈ C and λ ∈ [0, 1]. Four equivalent characterisations: (i) midpoint inequality (as stated, by definition); (ii) epigraph {(x, t) : t ≥…
A function f : C → R on a convex set C is convex iff f(λx + (1−λ)y) ≤ λf(x) + (1−λ)f(y) for all x, y ∈ C and λ ∈ [0, 1]. Four equivalent characterisations: (i) midpoint inequality (as stated, by definition); (ii) epigraph {(x, t) : t ≥…