Approximation Depth of Convex Polytopes
This addresses a theoretical problem in computational geometry, but it appears incremental as it builds on standard models and focuses on specific polytope properties.
The paper tackles the problem of approximating convex polytopes using Minkowski sums and unions, finding that simplices can only be trivially approximated and characterizing them as the only outer additive convex bodies.
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.