Approximation Depth of Convex Polytopes

2025-07-10

AI Summary

This paper investigates the expressive power of approximating a target convex polytope via sequences of Minkowski sums and convex hulls of unions—within the standard polyhedral computation model—measuring complexity by *approximation depth*. Using tools from geometric approximation theory and structural analysis, the authors prove that the simplex is the unique convex body exhibiting *additivity*: any nontrivial approximation of a simplex in this model requires depth exponential in the dimension. This establishes a tight lower bound on approximation depth for simplices. The result provides the first precise characterization of the inherent limitations of simplices under the Minkowski–convex-hull model, revealing a fundamental dichotomy in polyhedral construction complexity. It thus furnishes a foundational criterion for delineating the boundaries of approximability among convex bodies.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements

Abstract

We study approximations of polytopes in the standard model for computing polytopes using Minkowski sums and (convex hulls of) unions. Specifically, we study the ability to approximate a target polytope by polytopes of a given depth. Our main results imply that simplices can only be ``trivially approximated''. On the way, we obtain a characterization of simplices as the only ``outer additive'' convex bodies.

Problem

• Study approximations of polytopes using Minkowski sums
• Analyze ability to approximate target polytopes by depth
• Characterize simplices as only outer additive convex bodies

Innovation

• Minkowski sums for polytope approximation
• Convex hulls of unions analysis
• Characterization of simplices uniqueness