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.

翻译：$k$-一致超图的凸维数是指最小的维数$d$，使得存在一个将其顶点单射映射到$\mathbb{R}^d$中的方式，且所有超边的$k$-重心集处于凸位置。我们完全确定了完全$k$-一致超图的凸维数，这解决了Halman、Onn和Rothblum提出的一个公开问题，他们此前已解决了完全图的情形。我们还针对极值问题给出了上下界，该问题旨在估计具有凸维数$d$的$n$顶点$k$-一致超图的最大超边数量。为了证明这些结果，我们将问题重新表述为保持超单纯形顶点的仿射投影问题。更一般地，我们完整刻画了保持其$i$维骨架的投影。特别地，我们得到了线性van Kampen-Flores定理的超单纯形推广：对于每个$n$、$k$和$i$，我们确定了$(n,k)$-超单纯形在保持其$i$骨架的前提下可以线性投影到哪些维数。我们的结果可以直接用$k$-集和$(i,j)$-划分来解释，并且与在$k$个点集的Minkowski和中寻找大凸独立子集的问题密切相关。