Containment and inscribed simplices Daniel Klain 52A20convex geometrycontainmentcovering Let $K$ and $L$ be compact convex sets in $\mathbb{R}^n$. The following two statements are shown to be equivalent: \begin{enumerate} (i) For every polytope $Q \subseteq K$ having at most $n+1$ vertices, $L$ contains a translate of $Q$. (ii) $L$ contains a translate of $K$. \end{enumerate} Let $1 \leq d \leq n-1$. It is also shown that the following two statements are equivalent: \begin{enumerate} (i) For every polytope $Q \subseteq K$ having at most $d+1$ vertices, $L$ contains a translate of $Q$. (ii) For every $d$-dimensional subspace $\xi$, the orthogonal projection $L_{\xi}$ of the set $L$ contains a translate of the corresponding projection $K_{\xi}$ of the set $K$. \end{enumerate} It is then shown that, if $K$ is a compact convex set in $\mathbb{R}^n$ having at least $d+2$ exposed points, then there exists a compact convex set $L$ such that every $d$-dimensional orthogonal projection $L_{\xi}$ contains a translate of the projection $K_{\xi}$, while $L$ does not contain a translate of $K$. In particular, if $\dim K > d$, then there exists $L$ such that every $d$-dimensional projection $L_{\xi}$ contains a translate of the projection $K_{\xi}$, while $L$ does not contain a translate of $K$. Indiana University Mathematics Journal 2010 text pdf 10.1512/iumj.2010.59.4067 10.1512/iumj.2010.59.4067 en Indiana Univ. Math. J. 59 (2010) 1231 - 1244 state-of-the-art mathematics http://iumj.org/access/