Affine function

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Let LaTeX: f : X \rightarrow \mathbb{R}^n. Then, LaTeX: f is affine if LaTeX: X is a convex set and

f(\theta x + (1-\theta) y) = \theta f(x) + (1-\theta)f(y),

for all LaTeX: x, y \in X and LaTeX: \theta \in [0, 1].

Equivalently, LaTeX: f is affine if it is both convex and concave. Moreover, if LaTeX: X = \mathbb{R}^n, LaTeX: f is the translation of a linear function: LaTeX: a^T x + b.

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