Let X be a simplicial complex on vertex set V. We say that X is d-representable if it is isomorphic to the nerve of a family of convex sets in \({\mathbb {R}}^d\). We define the d-boxicity of X as the minimal k such that X can be written as the intersection of k d-representable simplicial complexes. This generalizes the notion of boxicity of a graph, defined by Roberts. A missing face of X is a set \(\tau \subset V\) such that \(\tau \notin X\) but \(\sigma \in X\) for any \(\sigma \subsetneq \tau \). We prove that the d-boxicity of a simplicial complex on n vertices without missing faces of dimension larger than d is at most \(\bigl \lfloor \left( {\begin{array}{c}n\\ d\end{array}}\right) /(d+1)\bigr \rfloor \). The bound is sharp: the d-boxicity of a simplicial complex whose set of missing faces form a Steiner \((d,d+1,n)\)-system is exactly \(\left( {\begin{array}{c}n\\ d\end{array}}\right) /(d+1)\). One of the main ingredients in the proof is the following bound on the representability of a complex: let \(V_1,\ldots , V_k\) be subsets of V such that \(V_i\notin X\) for all \(1\le i\le k\), and for any missing face \(\tau \) of X there is some \(1\le i\le k\) satisfying \(|\tau \setminus V_i|\le 1\). Then, X can be written as an intersection \(X=\bigcap _{i=1}^kX_i\), where, for all \(1\le i\le k\), \(X_i\) is a \((|V_i|-1)\)-representable complex. In particular, X is \(\bigl (\sum _{i=1}^k(|V_i|-1)\bigr )\)-representable.

令X是顶点集V上的单纯复形 。如果X与\({\mathbb {R}}^d\) 中的一组凸集的神经同构，我们说X是d 可表示的 。我们将X的d -boxicity定义为最小k，使得X可以写为k d -可表示单纯复形的交集。这概括了由 Roberts 定义的图的boxicity 的概念。X的缺失面是一个集合\(\tau \subset V\)使得\(\tau \notin X\)但\(\sigma \in X\) 对于任何\(\sigma \subsetneq \tau \)。我们证明了d的上单纯复-boxicity Ñ顶点而不错过尺寸大的面比d为至多\（\ bigl \ lfloor \左（{\开始{阵列} {C} n的\\ d \ {端数组}}\right) /(d+1)\bigr \rfloor \)。界限是尖锐的：一个简单复形的d -boxicity 其缺失的面集形成了 Steiner \((d,d+1,n)\) -system 正好是\(\left( {\begin{array}{ c}n\\ d\end{array}}\right) /(d+1)\)。证明中的主要成分之一是复数可表示性的以下界限：令\(V_1,\ldots , V_k\)是V 的子集使得\（V_I \ notin X \）对于所有\（1 \文件I \文件ķ\） ，和对于任何缺失面\（\ tau蛋白\）的X有一些\（1 \文件I \文件ķ\ )满足\(|\tau \setminus V_i|\le 1\)。那么，X可以写成一个交集\(X=\bigcap _{i=1}^kX_i\)，其中，对于所有\(1\le i\le k\)，\(X_i\)是一个\ ((|V_i|-1)\) - 可表示的复合体。特别是，X是\(\bigl (\sum _{i=1}^k(|V_i|-1)\bigr )\) -可表示的。