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!