Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 326 / 3 / 2 / b / Solution

Let $G=E\square F$. To prove convexity, take $u_1,u_2$ with finite $G$ values, $0<t<1$, and $\epsilon >0$. By the definition of infimum, choose $v_i$ withPut $u_t=t{u}_1+(1-t)u_2$ and $v_t=t{v}_1+(1-t)v_2$. Convexity of $E$ and $F$ givesLet $\epsilon \downarrow 0$. If either endpoint value is infinite, the convexity inequality is automatic, and $t=0,1$ gives equality. Thus the infimal convolution of two convex functions is convex under the stated properness assumption. The proof uses approximate minimizing splits, so no attainment of the inner infimum is assumed.