Fix an integer $n$, and consider the category $Conv$ of all convex subsets of $\mathbb{R}^n$, with morphisms given by inclusion. Given $A,B\in Obj(C)$, the Minkowski sum is defined as $A+B=\{a+b: a\in A, b\in B\}$.
Question: does $A+B$ satisfy a universal property in $Conv$?
If not, then is there an "intrinsic" definition of $A+B$? By "intrinsic", I mean that the definition can be expressed entirely in terms of operations on convex sets, and does not explicitly require quantification over points of $A$ and $B$. An example a definition I would consider "intrinsic" is $ConvHull(A)=\cap\{A': A\subset A', A' convex\}$.
(This is NOT a duplicate of categorical description of the Minkowski sum of polytopes because that question deals with a different categorical structure on the space of convex sets)