> **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!
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![EqTriCircum][1]
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**Answered** by Wlodzimierz Holsztynski in the comments: *Yes*, every $K$ has (many)
circumscribing regular simplices.
[1]: https://i.sstatic.net/pYXYu.jpg