# Hausdorff approximations and volume of tubes of singular algebraic sets

arXiv: https://arxiv.org/abs/2104.05053 · HTML: https://mlsanity.com/paper/2104.05053/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Apr 2021.

## Built on by (most cited first, 5 of 5)

- [Algebra Unveils Deep Learning -- An Invitation to Neuroalgebraic Geometry](https://mlsanity.com/paper/2501.18915/) (2025)
- [On the Geometry and Optimization of Polynomial Convolutional Networks](https://mlsanity.com/paper/2410.00722/) (2024)
- [Covering Number of Real Algebraic Varieties and Beyond: Improved Bounds and Applications](https://mlsanity.com/paper/2311.05116/) (2023)
- [Testing the variety hypothesis](https://mlsanity.com/paper/2507.16705/) (2025)
- [Tubular Neighbourhoods of Pfaffian Sets and Applications to Neural Networks](https://mlsanity.com/paper/2607.08370/) (2026)

## Abstract (arXiv)

We prove bounds for the volume of neighborhoods of algebraic sets, in the euclidean space or the sphere, in terms of the degree of the defining polynomials, the number of variables and the dimension of the algebraic set, without any smoothness assumption. This generalizes previous work of Lotz on smooth complete intersections in the euclidean space and of B"urgisser, Cucker and Lotz on hypersurfaces in the sphere, and gives a complete solution to Problem 17 in the book titled "Condition" by B"urgisser and Cucker.