Pome127837115DD3Lemma 19.7.11.127837115DD3 Let $D_ j\subseteq \mathbb {R}^ n$ $(j\geq 1)$ be a decreasing sequence of bounded convex sets. Assume that $\operatorname{vol}\bigcap _ j D_ j>0$. Then
Proof. The $\subseteq $ direction is clear. By convexity, it suffices to show that both sides have the same positive volume: indeed, a proper inclusion of closed convex sets with positive volume would cut off a non-empty relatively open convex piece by a separating hyperplane, hence would strictly decrease the volume. As the boundary of convex sets has zero Lebesgue measure, it follows that the volumes of both sides are equal to $\lim _{j\to \infty }\operatorname{vol}D_ j$. $\square$
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