Is it true that $\text{conv}(X)\times\text{conv}(Y)\subset \text{conv}(X\times Y)$, where $X,Y$ are subsets of a (not-necessarily-finite-dimensional) vector space? If the answer is “no”, what if we are in a Banach space?
This question provides a proof in finite-dimensional spaces, however, it seems to me that the proof works in any space (the invocation of Caratheodory is unnecessary). But this is such an elementary fact and proof that I would expect this to show up in textbooks, and I could not find any discussion of this.
