{
  "annotation_id": "1e5885f7-aac8-4d7e-bcbb-d353838fcad9",
  "date_created": "2026-02-17T05:53:24.155000Z",
  "date_modified": "2026-02-17T05:53:24.155000Z",
  "file_hash": "ce6c357c611066659846d27a59ef7f110de0dba6db33f87f80ac5e74498bc37d",
  "private": false,
  "record": {
    "abstract": "We study a family of non-local isoperimetric energies $E_{\\gamma,\\varepsilon}$ on the round sphere $M = S^n$, where the non-local interaction kernel $K_\\varepsilon$ is the fundamental solution of the Helmholtz operator $1 - \\varepsilon^2 \\Delta$. To analyse these energies, we introduce a Riemannian autocorrelation function $c_\\Omega$ associated to a measurable set $\\Omega\\subset M$, defined on any compact, connected, oriented Riemannian manifold without boundary $(M^n,g)$ of dimension $n\\ge2$. This function is intimately linked to Matheron\u0027s set covariogram from convex geometry. By establishing a characterisation of functions of bounded variation $BV(M)$ in terms of geodesic difference quotients, we show that $\\Omega$ has finite perimeter if and only if $c_\\Omega$ is Lipschitz, and we relate the Lipschitz constant to the perimeter of $\\Omega$. We show that on the round sphere $E_{\\gamma,\\varepsilon}$ admits a reformulation in terms of $c_\\Omega$, which allows us to compute the limit as $\\varepsilon \\to 0$ in a variational sense, that is, in the framework of $\\Gamma$-convergence.",
    "arxiv_id": "2601.10481",
    "authors": [
      "Michael Bleher",
      "Denis Brazke",
      "Sebastian Nill"
    ],
    "categories": [
      "math.AP",
      "math.DG"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "title": "A Riemannian Autocorrelation Function and its Application to Non-Local Isoperimetric Energies",
    "url": "https://arxiv.org/abs/2601.10481",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "cacb35a8-79ce-4ab9-9c1e-8d99da6b1bad",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}