# Approximation Depth of Convex Polytopes

arXiv: https://arxiv.org/abs/2507.07779 · HTML: https://mlsanity.com/paper/2507.07779/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 22 references that are in the map.
- First posted on arXiv 10 Jul 2025.

## Builds on (most cited first, 7 of 7)

- [Understanding Deep Neural Networks with Rectified Linear Units](https://mlsanity.com/paper/1611.01491/) (2016)
- [Towards Lower Bounds on the Depth of ReLU Neural Networks](https://mlsanity.com/paper/2105.14835/) (2021)
- [Lower Bounds on the Depth of Integral ReLU Neural Networks via Lattice Polytopes](https://mlsanity.com/paper/2302.12553/) (2023)
- [Counterexample-Guided Learning of Monotonic Neural Networks](https://mlsanity.com/paper/2006.08852/) (2020)
- [On Minimal Depth in Neural Networks](https://mlsanity.com/paper/2402.15315/) (2024)
- [On the Depth of Monotone ReLU Neural Networks and ICNNs](https://mlsanity.com/paper/2505.06169/) (2025)
- [Size and depth of monotone neural networks: interpolation and approximation](https://mlsanity.com/paper/2207.05275/) (2022)

## Abstract (arXiv)

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.