Submodular width is a central structural measure governing the complexity of conjunctive query evaluation. In this paper we recast submodular width in geometric terms. We how that submodular width can be approximated, up to a factor $3/2$, by a new branchwidth parameter defined in terms of edge separations in the hypergraph and the costs induced on them by admissible submodular functions. This reformulation turns lower bounds on submodular width into the problem of constructing well-balanced edge separations whose induced cost remains small. We then express this connection through a variational characterisation in terms of a convex body. Using these tools, we relate submodular width to more familiar graph-theoretic notions, including line-graph treewidth and multicommodity flow, and obtain general conditions under which submodular width is tightly linked to generalised hypertree width. In particular, under various natural conditions we show that \[ subw(H) \in Ω\left(\frac{ghw(H)}{\log ghw(H)} \right). \]

翻译：子模宽度是衡量合取查询评估复杂性的核心结构度量。本文从几何角度重新阐释子模宽度。我们证明，通过超图中边分离及其由可容许子模函数诱导的代价所定义的一种新分支宽度参数，可在$3/2$因子范围内逼近子模宽度。这一重构将子模宽度的下界问题转化为构造平衡边分离且诱导代价保持较小的问题。进而通过凸体的变分刻画建立该关联。借助这些工具，我们揭示了子模宽度与更经典的图论概念（包括线图树宽和多商品流）之间的关联，并获得了子模宽度与广义超树宽度紧密相关的通用条件。特别地，在多种自然条件下，我们证明了：\[ subw(H) \in Ω\left(\frac{ghw(H)}{\log ghw(H)} \right). \]