M-Representation of Polytopes

TL;DR

The paper introduces the M-representation of polytopes, enabling efficient computation of linear transformations, convex hulls, and Minkowski sums, with a more compact form for certain polytope combinations.

Contribution

It presents the M-representation and chain representation of polytopes, offering computational efficiency and smaller sizes compared to traditional representations.

Findings

Linear transformations, convex hulls, and Minkowski sums computed with linear complexity.

M-representation can be more compact than V-, H-, and Z-representations.

Chain representation allows direct computation of some operations, reducing conversion needs.

Abstract

We introduce the M-representation of polytopes, which makes it possible to compute linear transformations, convex hulls, and Minkowski sums with linear complexity in the dimension of the polytopes. When the polytope is a convex hull of a zonotope and a polytope, the representation size can be smaller than any of the known representations (V-representation, H-representation, and Z-representation). We also provide a variant of the M-representation: The chain representation is more compact and we can directly use it to compute linear transformations and convex hulls -- for all other operations on the chain representation, one requires a conversion to the M-representation.