intersection of two convex hulls

Question

Let $X=\{x_1,\ldots,x_p\}\subseteq\mathbb{R}^n$ and $Y=\{y_1,\ldots,y_q\}\subseteq\mathbb{R}^n$. Is there a method (by using some algorithm) to find $\mathrm{conv}(X)\cap \mathrm{conv}(Y)$ as $\mathrm{conv}(Z)$, where $Z\subseteq\mathbb{R}^n$?

Answer 1

Yes, you can. The magic words are Motzkin double description method.