Or simpler, just take any rectangular prism that is not a cube. Works in two dimensions too. Andy On Mon, May 27, 2019, 03:52 Dan Asimov <dasimov@earthlink.net> wrote:
There is one with 6 faces, anyhow.
(Far below if you want to try to find one without spoilage.)
—Dan
Let T be a skinny isosceles triangle with apex A.
Let the polyhedron P be the result of truncating [the product T x [0, 1] of T with the unit interval] by cutting off a neighborhood N of {A} x [0, 1], thereby creating a new face — a skinny rectangle R.
Then P has all 8 of its vertices on its circumsphere C. But clearly C would not touch the rectangle R if N is small enough.
—Dan
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