More generally, is every n-polytope with all its vertices on an n-sphere convex? Among the several images displayed on Wikipedia's circumscribed circle page, none depicts a non-convex polygon.
2026-03-28 08:08:45.1774685325
Is every cyclic polygon convex?
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Here's a non-convex example in three dimensions. Consider the union of two tetrahedra, each with base with vertices $(1,0,0)$, $(4/5,3/5,0)$ and $(4/5,-3/5,0)$. One of the tetrahedra has fourth vertex $(0,0,1)$ the other $(0,0,-1)$.
There's no example in the plane.