Factorizations over cones and their duals play central roles for many areas of mathematics and computer science. One of the reasons behind this is the ability to find a representation for various objects using a well-structured family of cones, where the representation is captured by the factorizations over these cones. Several major questions about factorizations over cones remain open even for such well-structured families of cones as non-negative orthants and positive semidefinite cones. Having said that, we possess a far better understanding of factorizations over non-negative orthants and positive semidefinite cones than over other families of cones. One of the key properties that led to this better understanding is the ability to normalize factorizations, i.e., to guarantee that the norms of the vectors involved in the factorizations are bounded in terms of an input and in terms of a constant dependent on the given cone. Our work aims at understanding which cones guarantee that factorizations over them can be normalized, and how this effects extension complexity of polytopes over such cones.

翻译：锥及其对偶上的分解在数学与计算机科学的诸多领域中扮演着核心角色。其原因之一在于，能够利用结构良好的锥族为各类对象找到一种表示，而这种表示正是通过这些锥上的分解来捕捉的。即使对于非负象限和半正定锥这类结构良好的锥族，关于锥上分解的若干重要问题仍未解决。尽管如此，相较于其他锥族，我们对非负象限和半正定锥上分解的理解要深入得多。促成这种更深入理解的关键特性之一，是能够对分解进行归一化，即确保分解中涉及的向量范数能够以输入量以及一个依赖于给定锥的常数为界。我们的工作旨在理解哪些锥能保证其上的分解可被归一化，以及这如何影响多面体在此类锥上的扩展复杂度。