# A simplex-based measure of symmetry

arXiv: https://arxiv.org/abs/2607.03815 · HTML: https://mlsanity.com/paper/2607.03815/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 24 references that are in the map.
- First posted on arXiv 4 Jul 2026.

## Builds on (most cited first, 8 of 8)

- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [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)
- [On the Expressiveness of Rational ReLU Neural Networks With Bounded Depth](https://mlsanity.com/paper/2502.06283/) (2025)
- [Better Neural Network Expressivity: Subdividing the Simplex](https://mlsanity.com/paper/2505.14338/) (2025)
- [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)

## Abstract (arXiv)

For compact convex sets L,K subset ℝn, denote by λK(L) the smallest size of a homothet of K that contains L. We define a measure of symmetry based on the n-simplex Delta = Deltan subset ℝn as the ratio [ ρ_Delta(L):=fracλ-Delta(L)λDelta(L). ] We study this measure and deduce the following results: (1) The classical Minkowski measure of symmetry m^*(L) can be defined as an affine-invariant version of ρ_Delta(L). (2) We improve the stability analysis for the Minkowski measure of symmetry; if m^*(L)≥ n-varepsilon then L is tfrac11-varepsilon-close to Delta in the Banach--Mazur distance. (3) We obtain a novel characterization of simplices as the only convex bodies K for which the function L mapsto λK(L) is additive (a property we term ``outer additivity''). (4) Motivated by the expressivity of ReLU neural networks, we study the depth complexity of polytopes in ℝn under the two operations: Minkowski sum and convex hull of a union. We prove the sharp bound ρ_Delta(P) ≤ 2d -1 for every polytope P of depth complexity d. In other words, simplices cannot be approximated by low-depth polytopes.