{
  "id": "2311.15152",
  "version": "v1",
  "published": "2023-11-26T00:53:05.000Z",
  "updated": "2023-11-26T00:53:05.000Z",
  "title": "How does the contraction property fail for convex functions on normed spaces?",
  "authors": [
    "Shin-ichi Ohta"
  ],
  "comment": "13 pages",
  "categories": [
    "math.OC"
  ],
  "abstract": "On Euclidean and Hilbert spaces, Riemannian manifolds, and CAT$(0)$-spaces, gradient flows of convex functions are known to satisfy the contraction property, which plays a fundamental role in optimization theory and possesses fruitful analytic and geometric applications. On (non-inner product) normed spaces, however, gradient flows of convex functions do not satisfy the contraction property. We give a detailed proof of this characterization of inner products, and discuss a possible form of a weaker contraction property on normed spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-26T00:53:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "convex functions",
      "contraction property fail",
      "normed spaces",
      "gradient flows",
      "weaker contraction property"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}