# A Riemannian Corollary of Helly's Theorem

arXiv: https://arxiv.org/abs/1804.10738 · HTML: https://mlsanity.com/paper/1804.10738/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Apr 2018.

## Built on by (most cited first, 2 of 2)

- [Riemannian Projection-free Online Learning](https://mlsanity.com/paper/2305.19349/) (2023)
- [Horospherical Depth and Busemann Median on Hadamard Manifolds](https://mlsanity.com/paper/2604.18242/) (2026)

## Abstract (arXiv)

We introduce a notion of halfspace for Hadamard manifolds that is natural in the context of convex optimization. For this notion of halfspace, we generalize a classic result of Gr"unbaum, which itself is a corollary of Helly's theorem. Namely, given a probability distribution on the manifold, there is a point for which all halfspaces based at this point have at least frac1n+1 of the mass. As an application, the gradient oracle complexity of convex optimization is polynomial in the parameters defining the problem.