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The Vandermonde determinant is used to prove that cyclic polytopes maximize the number of $i$-dimensional faces for each $i$ among all triangulations of a $(d-1)$-dimensional sphere having exactly n vertices. Specifically, the proof uses the result about the moment curve which David Hansen has given in his answer -- the The cyclic polytope $C(n,d)$ is the convex hull of any distinct $n$ distinct points on the moment curve {$(t,t^2,\dots ,t^d) | t\in {\mathbf R}$} $\subseteq {\mathbf R}^d$. I like the discussion of cyclic polytopes in G"unter Ziegler's book "Lectures on Polytopes". The wikipedia article en.wikipedia.org/wiki/Cyclic_polytope also seems to give a nice quick summary.

The Vandermonde determinant is used to prove that cyclic polytopes maximize the number of $i$-dimensional faces for each $i$ among all triangulations of a $(d-1)$-dimensional sphere having exactly n vertices. Specifically, the proof uses the result about the moment curve which David Hansen has given in his answer -- the cyclic polytope $C(n,d)$ is the convex hull of any distinct $n$ points on the moment curve {$(t,t^2,\dots ,t^d) | t\in {\mathbf R}$} $\subseteq {\mathbf R}^d$. I like the discussion of cyclic polytopes in G"unter Ziegler's book "Lectures on Polytopes". The wikipedia article en.wikipedia.org/wiki/Cyclic_polytope also seems to give a nice quick summary.