Convexity Preserved Under Affine Composition

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

Proof idea

Use $B(\lambda x+(1-\lambda)y)=\lambda B(x)+(1-\lambda)B(y)$ for $0\leq\lambda\leq1$ , then apply convexity of $f$ .