Does the concept of Minkowski Sums generalise naturally in infinite dimensional settings?

Question

I would like to use the concept of Minkowski Sums to study some convex analysis problems in an infinite dimensional setting, but all the papers and books I can find are referring to finite dimensional cases. Is there a reason for that? Are there any issues with generalising the concept to infinite dimensional Convex/Vector spaces?

Answer 1

No, Minkowski sums arise most naturally in infinite dimensional settings, especially in topological vector spaces, Let me illustrate:

-Many familiar properties still hold in infinite dimensional spaces. For example, the sum of convex is convex. In a topological vector space, the sum of a compact set and a closed set is closed. The sum of an open set and a some other set is open. The sum of compact sets is compact, etc. The most fundamental fact used is the fact that the function $a: X \times X \rightarrow X$ given by addition: $a(x,y) = x+y$. is continues.

-Since in topological vector spaces we don't necessarily have a metric, once resorts to "taking a sum with open sets" to express uniform statements. For example: in a metric space $X$ you can say that if $C,K \subset X$, the first closed, the latter compact and are disjoint, one can separate them by a $\epsilon-$neighborhood, meaning $\{x \in X \vert d(x,K) < \epsilon\}$ is disjoint from $C$. In a topological vector space $X$ you can instead formulate it by saying there exists a neighborhood $0 \in U$ of $0$ such that $K+U \cap C+U = \varnothing$. Which expresses a certain "uniform" separation.

These are just some reasons, but there is no obstruction in talking about sums of sets in infinite dimensional spaces. In fact it is done in practice, it just maybe used in slightly different contexts than the ones you know from convex analysis in finite dimensional space.