I'd like to have the answer to the following question. If $X_1,X_2\subseteq \mathbb{R}^n$ are convex and compact sets of dimension $n$, does the following hold:
$$\mathrm{\mathop{int}}(X_1+X_2)\subseteq\mathrm{\mathop{int}}(X_1)+\mathrm{\mathop{int}}(X_2)$$
? I expect the equation above to be true, but I don't have a proof yet. The case I'm struggling with is, when we suppose a $y\in\mathrm{\mathop{int}}( X_1+X_2)$ has a decomposition $y=x_1+x_2$, where $x_i\in\mathrm{\mathop{bd}} (X_i)$. I checked the following link:
Relative interior of the sum of two convex sets
It suggests, to use a rpresentation $y=\lambda y_1+(1-\lambda)y_2$, where $y_i\in X_1+X_2$ and $\lambda\in(0,1)$. Then we may write $y_i=x_1^{(i)}+x_2^{(i)}$, where $x_j^{(i)}\in X_j$, and we find
$$y= \left(\lambda x_1^{(1)}+(1-\lambda)x_1^{(2)}\right)+\left(\lambda x_2^{(1)}+(1-\lambda)x_2^{(2)}\right).$$
He proceeds to argue, that $\lambda x_j^{(1)}+(1-\lambda)x_j^{(2)}\in\mathrm{\mathop{int}}(X_j)$, but how do we know, that the line $\left[x_j^{(1)},x_j^{(2)}\right]$ is not contained in the boundary?
NO it does not hold in $\Bbb R ^2$ take $X_1=0$ and $X_2 = \Bbb B$ closed Unit ball. then $\rm RHS = \emptyset$
It does hold if $\rm int ~ X_i \neq \emptyset , ~~ i=1,2$ .