Let $A_1,\dots,A_k \subset \mathbb{R}^n$. If $x \in \text{conv}(A_1+…+A_k)$, then $x \in \text{conv}(A_1’+…+A_k’)$, with finite $A_i’ \subset A_i$.

Question

Let $A_1,\dots,A_k \subset \mathbb{R}^n$. If $x \in \text{conv}(A_1+…+A_k)$, then $x \in \text{conv}(A_1’+…+A_k’)$, with finite $A_i’ \subset A_i$.

I’m trying to verify this remark. If $x$ can be bounded by a simplex in $\text{conv}(A_1+…+A_k)$, then we can use it to find such $A_1’,…,A_k’$ where $x$ has a representation in $\text{conv}(A_1’+…+A_k’)$. But what do I do when $x$ cannot be bounded that way? For example, take $A_1,A_2$ to be a ball and a rectangle in $\mathbb{R}^2$, and $x$ a corner in $\text{conv}(A_1+A_2)$.

Answer 1

The convex hull of a set $A$ is equivalent to the set of all convex combinations of points in $A$ (see for example Convex hull: Equivalence of definitions).

Therefore, if $x \in \operatorname{conv}(A_1+ A_2 + \cdots +A_k)$ then $$ x = \sum_{j=1}^N \lambda_j (a_{j, 1} + a_{j, 2} + \cdots + a_{j, k}) $$ where $\lambda_k \ge 0$ are real numbers with $\sum_{k=1}^N \lambda_k = 1$, and $a_{j, i} \in A_i$. Then $$ A_i' = \{ a_{1, i}, a_{2, i}, \ldots, a_{N, i} \} $$ are finite subsets of $A_i$, and $x \in \operatorname{conv}(A_1'+ A_2' + \cdots +A_k')$.