This is Problem 263 in Theorems and Problems in Functional Analysis.
Let $A$ be any bounded set in the plane and $B$ the unit disk $x^2+y^2<1$. Prove that for any positive numbers $\alpha$ and $\beta$ the set $\alpha A+\beta B$ is Lebesgue-measurable, and if $A$ is convex, then $$\mu(\alpha A+\beta B) = S\alpha^2 + L\alpha\beta + \pi\beta^2.$$ What is the meaning of the coefficients $S$ and $L$?
The hint tells that $S=\mu(A)$ and $L$ is the perimeter of $A$. I have no idea how to prove it, but I think the conclusion can be extended to higher dimensions. Thanks for any help!