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Sep 6, 2014 at 17:38 comment added Joseph O'Rourke Atsushi (above) proved: There is a simplex $S$ with Vol$(S) \le \;d^{d-1}$Vol$(K)$.
Sep 6, 2014 at 13:32 comment added Joseph O'Rourke @PietroMajer: I think what's missing here is that indeed the polar dual w.r.t. the simplex centroid maps to a simplex inscribed in the dual shape, but not all convex bodies can be obtained through this duality.
Sep 6, 2014 at 11:58 history edited Joseph O'Rourke CC BY-SA 3.0
Answered notice.
Sep 6, 2014 at 7:55 comment added Pietro Majer Precisely: it is not obvious (to me) that a simplex growing inside $L$ can in fact keep growing till all its vertices have reached the boundary of $L$. But the positive answer to Q implies a positive answer to Q' by duality, right?
Sep 6, 2014 at 7:49 comment added Pietro Majer Note the dual question: Q' Does every compact convex body $L$ in $\mathbb{R}^d$ admit an inscribed regular simplex, each vertex of which lies in the boundary of $L$ ? While Q admits the simple construction by moving planes described above, it is not obvious to me how the dual construction should work for the answer of Q'.
Sep 6, 2014 at 4:43 comment added Atsushi Kanazawa 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.
Sep 6, 2014 at 2:43 comment added Joseph O'Rourke @WlodzimierzHolsztynski: I like to believe that Thales would be pleased to be cited in this context, 2,500 years later. :-)
Sep 6, 2014 at 2:40 comment added Włodzimierz Holsztyński Thank you Joseph. Indeed, in $\ \mathbb R^n,\ $ when one vertex is fixed, and we move the opposite opposite plane then according to Thales we get a homothetic simplex (hence regular if we started with a regular simplex).
Sep 6, 2014 at 2:29 comment added Joseph O'Rourke @WlodzimierzHolsztynski: "the simplex stays regular": Very nice!!
Sep 6, 2014 at 2:23 comment added José Hdz. Stgo. I've just edited the link...
Sep 6, 2014 at 2:21 comment added José Hdz. Stgo. 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
Sep 6, 2014 at 2:19 history edited Joseph O'Rourke CC BY-SA 3.0
added 108 characters in body
Sep 6, 2014 at 2:17 comment added Włodzimierz Holsztyński 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).
Sep 6, 2014 at 2:06 comment added Włodzimierz Holsztyński 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.
Sep 6, 2014 at 2:00 history asked Joseph O'Rourke CC BY-SA 3.0