The convex dimension of hypergraphs and the hypersimplicial Van Kampen-Flores Theorem

The convex dimension of a $k$-uniform hypergraph is the smallest dimension $d$ for which there is an injective mapping of its vertices into $\mathbb{R}^d$ such that the set of $k$-barycenters of all hyperedges is in convex position. We completely determine the convex dimension of complete $k$-uniform hypergraphs, which settles an open question by Halman, Onn and Rothblum, who solved the problem for complete graphs. We also provide lower and upper bounds for the extremal problem of estimating the maximal number of hyperedges of $k$-uniform hypergraphs on $n$ vertices with convex dimension $d$. To prove these results, we restate them in terms of affine projections that preserve the vertices of the hypersimplex. More generally, we provide a full characterization of the projections that preserve its $i$-dimensional skeleton. In particular, we obtain a hypersimplicial generalization of the linear van Kampen-Flores theorem: for each $n$, $k$ and $i$ we determine onto which dimensions can the $(n,k)$-hypersimplex be linearly projected while preserving its $i$-skeleton. Our results have direct interpretations in terms of $k$-sets and $(i,j)$-partitions, and are closely related to the problem of finding large convexly independent subsets in Minkowski sums of $k$ point sets.