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.

k一致超图的凸维是最小维d，对其的顶点有一个射影映射到${\mathbb{[R}}^d$这样所有超边的k -barycenters的集合处于凸位置。

我们完全确定完整的k一致超图的凸维，这解决了Halman，Onn和Rothblum提出的一个开放问题，他们解决了完整图的问题。我们还为估计在凸维为d的n个顶点上的k一致超图的最大超边数的极值问题提供了上下界。

为了证明这些结果，我们根据仿射投影来重述它们，该仿射投影保留了超单纯形的顶点。更一般地，我们提供保留其i维骨架的投影的完整特征。特别是，我们获得了线性van Kampen-Flores定理的超简单泛化：对于每个n，k和i，我们都可以确定$（ñ，ķ）$-hypersimplex在保留其i-骨架的同时进行线性投影。

我们的结果对k集和$（\mathrm{一世}，Ĵ）$-分区，并且与在k个点集的Minkowski和中找到大的凸独立子集的问题密切相关。