  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.
