> **Q**.
Does every (perhaps smooth) compact convex body $K$ in $\mathbb{R}^d$ admit a
circumscribing simplex, each facet of which touches (shares a point with) $K$?
How about a circumscribing *regular* simplex?

This question is inspired by this result (in a paper I have yet to access):

> Shizuo Kakutani. 
"A proof that there exists a circumscribing cube around any bounded
closed convex set in $\mathbb{R}^3$."
*Ann. Math.*, 43(4):739–741, 1942. 

Perhaps the regular simplex question is answered already in $\mathbb{R}^2$?
If so, I would appreciate a reference. Thanks!