Re: Closed convex hull and cube

From: David Eppstein (eppstein_at_ics.uci.edu)
Date: 02/10/05 

Date: Thu, 10 Feb 2005 15:27:08 -0800

In article <slrnd0nnan.p80.tim-via-n.i.net@soprano.little-possums.net>,
 Timothy Little <tim-via-n.i.net@little-possums.net> wrote:

> "Gerry Myerson" <gerry@maths.mq.edi.ai.i2u4email> wrote:
> > Take a cylinder.
> > Shear it so its cross-sections are ellipses and
> > its axis is not orthogonal to its base.
> > Can you inscribe a cube in that?
>
> If I'm reading it correctly, yes.
>
> If we rotate it so that its axis is vertical, then it is just a
> squashed cyclinder with elliptical cross-sections in the horizontal
> planes. Inscribe a square in the cross-section, and extend it
> vertically to a cube.
>
> Too much symmetry :)

I've been trying to think of other convex bodies that don't have eight
cube vertices on their boundary, but it's difficult. E.g. truncating
one corner of a cube still allows an inscribed cube on the opposite
corner, and truncating two opposite corners allows a twisted cube inside
similar to the ones described in
<http://www.ics.uci.edu/~eppstein/junkyard/box-in-box.html>.

My best candidate so far: take three unit vectors at slightly less than
90 degree angles to each other and generate a parallelepiped from them
(i.e. the convex hull of the vector sums of all eight possible subsets
of the vectors). Does this have an inscribed cube? 

-- 
David Eppstein
Computer Science Dept., Univ. of California, Irvine
http://www.ics.uci.edu/~eppstein/

Relevant Pages

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