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Approximation Depth of Convex Polytopes

arXiv 2507.07779 ↗First posted 10 Jul 2025math.MG cs.CG cs.LGcited by 0

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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.

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.

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7 of 22 references.

References · 7

  1. 2016Understanding Deep Neural Networks with Rectified Linear Unitscited by 257
  2. 2021Towards Lower Bounds on the Depth of ReLU Neural Networkscited by 56
  3. 2023Lower Bounds on the Depth of Integral ReLU Neural Networks via Lattice Polytopescited by 36
  4. 2020Counterexample-Guided Learning of Monotonic Neural Networkscited by 34
  5. 2024On Minimal Depth in Neural Networkscited by 11
  6. 2025On the Depth of Monotone ReLU Neural Networks and ICNNscited by 11
  7. 2022Size and depth of monotone neural networks: interpolation and approximationcited by 10

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