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Supremum of Convex Functions

The pointwise supremum of any family of convex functions is convex

Supremum of Convex Functions: Let XX be a and let {fi:X→R‾}i∈I\{f_i:X\to\overline{\mathbb{R}}\}_{i\in I} be a family of indexed by a nonempty set II. Define

f(x):=sup⁡i∈Ifi(x).f(x):=\sup_{i\in I} f_i(x).

Then ff is convex on XX.

Remarks

This extends the "finite maximum" case in and is used heavily to construct convex envelopes and support functions.

Proof sketch (idea): For each ii, Jensen gives fi(λx+(1−λ)y)≤λfi(x)+(1−λ)fi(y)≤λf(x)+(1−λ)f(y)f_i(\lambda x+(1-\lambda)y)\le \lambda f_i(x)+(1-\lambda)f_i(y)\le \lambda f(x)+(1-\lambda)f(y). Taking sup⁡i\sup_i on the left yields Jensen for ff.