Convex Components and Multi-Slice Decompositions via Convex Functions

This paper develops a comprehensive theory of multi-slice decompositions via convex functions, extending the classical framework of slices determined by linear functionals to arbitrary convex functions with disjoint zero sets. We establish a fundamental structure theorem that completely characterizes the convex component decomposition of multi-slices, showing that under natural conditions of pairwise disjoint zero sets and convex separation, the multi-slice decomposes canonically into convex components that correspond precisely to the individual functions in the family. Our results reveal several key properties: the component-wise exposing nature of the supremum function, the closedness of components in appropriate topologies, the maximality of the resulting decomposition, and the affine invariance of convex component structures under injective affine maps. These contributions significantly extend the existing theory of multi-slices and convex components, providing new tools for understanding the geometric structure of convex sets under nonlinear constraints, with potential applications in optimization theory, high-dimensional data analysis, and modern convex geometry.