# Extension complexities of Cartesian products involving a pyramid

## TL;DR

This paper proves that for the Cartesian product of two polytopes, if one is a pyramid, the extension complexity of the product equals the sum of the individual complexities, resolving an open question in polytope theory.

## Contribution

It establishes that the extension complexity of the Cartesian product of a pyramid and another polytope is additive, confirming the conjecture in this specific case.

## Key findings

- Extension complexity of the product equals sum when one polytope is a pyramid.
- Addresses an open question in polytope extension complexity.
- Provides a new understanding of Cartesian products involving pyramids.

## Abstract

It is an open question whether the linear extension complexity of the Cartesian product of two polytopes P, Q is the sum of the extension complexities of P and Q. We give an affirmative answer to this question for the case that one of the two polytopes is a pyramid.