# Universal Barrier is n-Self-Concordant

arXiv: https://arxiv.org/abs/1809.03011 · HTML: https://mlsanity.com/paper/1809.03011/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Sep 2018.

## Built on by (most cited first, 6 of 6)

- [Sampling with Riemannian Hamiltonian Monte Carlo in a Constrained Space](https://mlsanity.com/paper/2202.01908/) (2022)
- [Best-of-Three-Worlds Linear Bandit Algorithm with Variance-Adaptive Regret Bounds](https://mlsanity.com/paper/2302.12370/) (2023)
- [The entropic barrier is n-self-concordant](https://mlsanity.com/paper/2112.10947/) (2021)
- [Beyond the d2.5-mixing bound for Dikin walks on polytopes](https://mlsanity.com/paper/2607.13943/) (2026)
- [Self-Concordant Perturbations for Linear Bandits](https://mlsanity.com/paper/2510.24187/) (2025)
- [Decomposable Non-Smooth Convex Optimization with Nearly-Linear Gradient Oracle Complexity](https://mlsanity.com/paper/2208.03811/) (2022)

## Abstract (arXiv)

This paper shows that the self-concordance parameter of the universal barrier on any n-dimensional proper convex domain is upper bounded by n. This bound is tight and improves the previous O(n) bound by Nesterov and Nemirovski. The key to our main result is a pair of new, sharp moment inequalities for s-concave distributions, which could be of independent interest.