Convexity Preserved Under Affine Composition

Affine Composition Rule: Let $B:X\to Y$ be an affine mapping between vector spaces , and let $f:Y\to\overline{\mathbb{R}}$ be a convex function . Then $f\circ B$ is convex on $X$.

This is the basic “change of variables” principle in convex analysis: restricting a convex function to an affine subspace or composing with an affine parameterization preserves convexity.

Proof sketch (idea): Use the defining identity for affine maps, $B(\lambda x+(1-\lambda)y)=\lambda B(x)+(1-\lambda)B(y)$, and then apply Jensen’s inequality for $f$.