Let (n \geq 2) be an integer. The regular (n)-gon inscribed in the complex unit circle and having (1) as a vertex, a.k.a., the convex hull of the (n)th roots of unity, is closed under complex multiplication.
The set of (n)th roots of unity is closed under multiplication, and
The product of two convex linear combinations of the vertices is itself a convex linear combination.
It's crucial to take the "standard" (n)-gon whose vertices are the (n)th roots of unity, i.e., not to take an arbitrary regular (n)-gon inscribed in the unit circle. The animations show the situation for (n = 7), with roots of unity ("standard") on the left, and the polygon rotated by one-tenth of a radian ("non-standard") on the right.
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