{
  "abstract": "We establish an $Ω(d)$ lower bound on the competitive ratio of convex body chasing in $\\mathbb{R}^d$ with movement costs measured in any norm. Previously, the best known lower bound valid across all norms was $Ω(d^{1/6})$, established by Papazov (2023); stronger lower bounds were known for $\\ell_p$ norms, where Bubeck, Klartag, Lee, Li, and Sellke (2019) had established a lower bound of $Ω(\\max\\{\\sqrt{d}, d^{1-1/p}\\})$. In particular, a linear lower bound was known only for the $\\ell_\\infty$ norm, realized by the standard construction of chasing faces of a hypercube. Our lower bound strictly improves on the prior universal and $\\ell_p$ lower bounds for $p < \\infty$ and shows that the functional Steiner point algorithm of Sellke (2023) - which has competitive ratio $O(d)$ - is asymptotically optimal for every norm. Our approach is based on the recursive construction of Bubeck, Coester, and Rabani (2023) for metrical service systems, which gives a lower bound of $Ω(r^2)$ for requests of size $O(r)$. We show that requests in their construction, each comprising a set of discrete points in a graph metric, can be mapped to the vertices of convex requests in a $(2r+1)$-dimensional $\\ell_p$ vector space while approximately preserving movement costs up to a factor of $O(r)$; the general result then follows via Milman's (1985) quotient-of-a-subspace theorem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.32670",
  "author_total": 1,
  "authors": [
    {
      "name": "Nicolas Christianson"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ex5cb6wqscepc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.32670",
  "fulltext_url": null,
  "id": "ex5cb6wqscepc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.32670",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.CG"
    ],
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "1705|Optimization and Search Problems"
    ]
  ],
  "openalex": "W7223700116",
  "pdf_url": "https://arxiv.org/pdf/2610.32670",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ex5cb6wqscepc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2610.32670",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.32670"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A Linear Lower Bound for Convex Body Chasing in Every $\\ell_p$ Norm",
  "title_text": "A Linear Lower Bound for Convex Body Chasing in Every ℓ_p Norm",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:38Z",
  "url": "https://research.draconitic.com/paper/ex5cb6wqscepc",
  "venue": "arXiv",
  "versions": []
}