# A Nearly-Linear Bound for Chasing Nested Convex Bodies

arXiv: https://arxiv.org/abs/1806.08865 · HTML: https://mlsanity.com/paper/1806.08865/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Jun 2018.

## Built on by (most cited first, 9 of 9)

- [Optimal Algorithms for Online Convex Optimization with Adversarial Constraints](https://mlsanity.com/paper/2310.18955/) (2023)
- [Revisiting Smoothed Online Learning](https://mlsanity.com/paper/2102.06933/) (2021)
- [Chasing Convex Bodies and Functions with Black-Box Advice](https://mlsanity.com/paper/2206.11780/) (2022)
- [Smoothed Online Convex Optimization Based on Discounted-Normal-Predictor](https://mlsanity.com/paper/2205.00741/) (2022)
- [Convex Optimization with Nested Evolving Feasible Sets](https://mlsanity.com/paper/2605.07386/) (2026)
- [Nested Convex-Body Chasing for Online Optimization with Evolving Feasible Sets](https://mlsanity.com/paper/2608.29074/) (2026)
- [Stable Movement for Nondual Lipschitz Convex Optimization: Efficiency and Nearly Optimal Oracle Rates](https://mlsanity.com/paper/2609.20701/) (2026)
- [Consistent Model Chasing Is Minimax Optimal: The Exact Value of Scalar Adversarial Adaptive Control under Large Parametric Uncertainty](https://mlsanity.com/paper/2608.13651/) (2026)
- [Convex Optimization with Nested Evolving Feasible Sets (CONES) under Time-Varying Loss Functions](https://mlsanity.com/paper/2609.11207/) (2026)

## Abstract (arXiv)

Friedman and Linial introduced the convex body chasing problem to explore the interplay between geometry and competitive ratio in metrical task systems. In convex body chasing, at each time step t ∈ mathbbN, the online algorithm receives a request in the form of a convex body Kt subseteq ℝd and must output a point xt ∈ Kt. The goal is to minimize the total movement between consecutive output points, where the distance is measured in some given norm. This problem is still far from being understood, and recently Bansal et al. gave an algorithm for the nested version, where each convex body is contained within the previous one. We propose a different strategy which is O(d log d)-competitive algorithm for this nested convex body chasing problem, improving substantially over previous work. Our algorithm works for any norm. This result is almost tight, given an Omega(d) lower bound for the ℓ∞.