{
  "abstract": "We extend the recent work of Reis and Rothvoss on sparsifying sums of $\\ell_1$ norms to the more general task of sparsifying (Minkowski) sums of centrally symmetric, convex sets. As our main result, we prove that for any $\\varepsilon > 0$ and centrally symmetric, convex sets $C_1, \\ldots, C_m\\subseteq\\mathbb{R}^n$ there is a choice of weights $λ_1, \\dots , λ_m \\in \\mathbb{R}_{\\geq 0}$ such that at most $O(n / \\varepsilon^2)$ of the weights are non-zero, and \\[(1 - \\varepsilon)\\cdot C\\subseteq\\sum_{i = 1}^mλ_i\\cdot C_i\\subseteq(1 + \\varepsilon)\\cdot C,\\] where $C:= C_1 + \\cdots + C_m$ refers to the Minkowski sums of the sets $C_1, \\ldots, C_m$, and $λ\\cdot C$ refers to the dilation of the set $C$. As immediate applications of this result, we obtain sparsifiers of size $O(n / \\varepsilon^2)$ for sparsifying sums of seminorms in $n$-dimensional space, improving on the $O\\left ( \\frac{n \\log(n/\\varepsilon) \\cdot \\log^{2.5}(n)}{\\varepsilon^2} \\right )$ size sparsifiers from the work of Jambulapati, Lee, Liu, and Sidford (FOCS 2023). This further yields optimal size hypergraph cut sparsifiers with $O(n / \\varepsilon^2)$ hyperedges, improving on the $O(n \\log(n) / \\varepsilon^2)$ size sparsifiers from the work of Chen, Khanna, and Nagda (FOCS 2020). More generally, this also gives optimal size sparsifiers for sums of symmetric submodular functions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20238",
  "author_total": 6,
  "authors": [
    {
      "name": "Arpon Basu"
    },
    {
      "name": "Joshua Brakensiek"
    },
    {
      "name": "Yeyuan Chen"
    },
    {
      "name": "Aaron Putterman"
    },
    {
      "name": "Victor Reis"
    },
    {
      "name": "Zihan Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cce8w9ptt1g5q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20238",
  "fulltext_url": null,
  "id": "cce8w9ptt1g5q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20238",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "1703|Complexity and Algorithms in Graphs"
    ]
  ],
  "openalex": "W7213580772",
  "pdf_url": "https://arxiv.org/pdf/2609.20238",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cce8w9ptt1g5q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.20238",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20238"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Optimal Sparsifiers for Minkowski Sums and Sums of Seminorms",
  "title_text": "Optimal Sparsifiers for Minkowski Sums and Sums of Seminorms",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/cce8w9ptt1g5q",
  "venue": "arXiv",
  "versions": []
}