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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!


      EqTriCircum


Answered by Wlodzimierz Holsztynski in the comments: Yes, every $K$ has (many) circumscribing regular simplices.

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    $\begingroup$ In $\ \mathbb R^2\ $ just move the lines till they touch the convex body. You can position your regular simplex (equilateral triangle) in every possible direction. $\endgroup$ – Włodzimierz Holsztyński Sep 6 '14 at 2:06
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    $\begingroup$ The same must be true in dimension $n$. When you move faces (each independently) of a regular simplex in parallel, the simplex stays regular (just make sure that the intersection of the chosen open half-spaces stays non-empty). $\endgroup$ – Włodzimierz Holsztyński Sep 6 '14 at 2:17
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    $\begingroup$ You may want to take a look at the references mentioned by Prof. Hatcher in his answer to the following question of mine: mathoverflow.net/questions/26318/points-on-a-sphere $\endgroup$ – José Hdz. Stgo. Sep 6 '14 at 2:21
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    $\begingroup$ I've just edited the link... $\endgroup$ – José Hdz. Stgo. Sep 6 '14 at 2:23
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    $\begingroup$ I gave a bound for the volume of such a simplex in link.springer.com/article/10.1007%2Fs00013-014-0616-6 It is interesting to ask about regular simplex case. $\endgroup$ – Atsushi Kanazawa Sep 6 '14 at 4:43

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