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Dima Pasechnik
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Do highly symmetric cones have "small" supporting hyperplanes?

Let $C$ be a full-dimensional cone in $\mathbb{R}^{d}$, defined as the positive span of $c = {n \choose 3} \gg d$ vectors. $C$ is highly symmetric in the following sense: each such vector is labelled with a distinct unordered triple from $[n]$, and the automorphism group of $C$ is basically $S_n$: for each permutation $\pi : [n] \to [n]$, there is a linear transformation of $\mathbb{R}^d$ that takes each vector labelled $(x, y, z)$ to the vector $(\pi(x), \pi(y), \pi(z))$.

My intuition is that any supporting hyperplane to $C$ can have only $O(d/n)$ of these vectors, roughly because the symmetry constraints prevent too many supporting vectors from being concentrated on one side of the cone. Is this true?