Suppose $S\subset{\mathbb R}^n$ is an infinite subset that is in *general position* which means that the intersection of $S$ with every affine subspace of dimension $d<n$ always contains at most $(d+1)$ points. Are there
two distinct points $a,b\in S$ such that $(a+b)/2$ is contained
in the interior of the convex hull of $S$ ?

## 1 Answer

If $n\geq 4$, let $S$ be a moment curve $f(t)=(t,t^2,\dots,t^n),t\in \mathbb{R}$. Any hyperplane contains at most $n$ points from $S$, since polynomial of degree at most $n$ has at most $n$ roots. So, any $n+1$ points of $S$ are in general position, hence $S$ is in general position. For any two points $a=f(u),b=f(w)$ the polynomial $(t-u)^2(t-v)^2=c_0+c_1t+c_2t^2+c_3t^3+c_4t^4$ is non-negative, hence the hyperplane $H$ defined by equation $c_0+c_1x_1+c_2x_2+c_3x_3+c_4x_4=0$ contains both points $a,b$, and the whole $S$ lies on the same side of $H$. It follows that $(a+b)/2$ does not lie in the interior of the convex hull of $S$. This is known as cyclic polytope.

If dimension is at most 3, the answer is positive even if $|S|=n+1$.