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  "slug": "riesz-capacity-ratio-conjectures",
  "title": "Sharp ratios of negative-exponent Riesz capacities",
  "shortTitle": "When is the regular simplex the best shape? Sharp ratios of Riesz capacities",
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  "oneLine": "Two points maximise one ratio of negative-exponent Riesz capacities on every compact metric space, the regular simplex maximises the reverse ratio in Euclidean space when the smaller exponent is at least 2, and that threshold is exact: below 2, splitting one simplex vertex wins.",
  "abstract": "Choose two random points of a compact set independently from the same distribution, and choose the distribution to make the average of the r-th power of their distance as large as possible; call that maximum M_r and put C_r = M_r^(1/r). For r = 2 this is twice the largest variance, for r = 1 the largest mean distance, and for Euclidean sets C_r is the Riesz capacity with exponent −r. Ratios of two such capacities measure shape, not size, and Clark and Laugesen (2024/2025) conjectured which shapes make them extreme. With 0 < s < t and γ = 1/s − 1/t, this candidate proves C_t/C_s ≤ 2^γ on every compact metric space, with two points extremal (the same short argument was posted publicly in a GitHub issue on 27 October 2026), and C_s/C_t ≤ (n/(n+1))^γ for compact sets in R^n when 2 ≤ s < t, with the regular simplex extremal (much of this range follows from Davies, Lim and McCann 2022; the paper makes the overlap exact and identifies a wedge of exponents near s = 2, for n ≥ 3, where only this argument applies). It then shows that exponent two is an exact threshold: for n ≥ 2 and s < 2, splitting one simplex vertex into two nearby points beats the simplex, so the regular simplex is optimal if and only if s ≥ 2. It also gives the sharp three-point lower bound for 0 < s < 1 and stability results with optimal exponents 1/s, 1/4 and 1/2; at s = 2 the simplex bound is a finite-exponent form of Jung's theorem. AI-assisted, unrefereed candidate; no external peer review, independent reproduction or formal verification is claimed.",
  "datePublished": "2026-11-03",
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  "keywords": [
    "Riesz capacity",
    "Riesz energy",
    "capacity ratio",
    "regular simplex",
    "Jung theorem",
    "metric spaces",
    "equilibrium measures",
    "stability"
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  "keyResults": [
    "Two-point bound (Theorem 1, proof): C_t/C_s ≤ 2^γ, γ = 1/s − 1/t, on every compact metric space, with equality iff M_s = D^s/2; the same two-step argument was posted publicly on 27 October 2026 (GitHub issue #22668) and is not claimed new.",
    "Simplex bound (Theorem 2 and Corollary 4, proof): C_s/C_t ≤ (n/(n+1))^γ for compact K ⊂ R^n and 2 ≤ s < t; at s = 2 this is M_t ≥ c_n^{1−t/2} 2^{t/2} R^t in terms of the circumradius, a finite-exponent form of Jung's theorem.",
    "Overlap with Davies–Lim–McCann (Proposition 5 and Example 6, proof): Theorem 2 at (s, t) follows from simplex minimality of their attractive–repulsive energy, which fails on a wedge W_n of exponents (n ≥ 3) where Theorem 2 still holds; the uniform measure on the sphere of R^3 is an explicit witness.",
    "Exact threshold (Theorem 7 and Corollary 8, proof): for n ≥ 2, 0 < s < 2 and t > s, splitting one vertex of a regular simplex increases C_s/C_t, so the regular simplex maximises C_s/C_t if and only if s ≥ 2.",
    "Sharp stability (Theorems 11, 13 and 14 and Corollary 12, proof): for 0 < s < 1, M_s/D^s ≥ Φ_s(ρ/D) in terms of the Hausdorff distance ρ to a diametral pair, attained by three-point spaces; near-extremal sets are close to two points or a simplex with optimal deficit exponents 1/s, 1/4 (at s = 1) and 1/2; constants explicit but not optimised."
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  "relatedWorks": [
    {
      "citation": "Clark and Laugesen (2025). Maximizing Riesz Capacity Ratios: Conjectures and Theorems. SIAM Journal on Mathematical Analysis. Poses the two-point and regular-simplex conjectures settled here and proves the planar cases.",
      "url": "https://doi.org/10.1137/24M171992X",
      "doi": "10.1137/24M171992X"
    },
    {
      "citation": "Clark and Laugesen (2025). Riesz capacity: monotonicity, continuity, diameter and volume. Analysis and Mathematical Physics. Monotonicity and diameter facts refined in Section 5.",
      "url": "https://doi.org/10.1007/s13324-024-01000-2",
      "doi": "10.1007/s13324-024-01000-2"
    },
    {
      "citation": "Davies, Lim and McCann (2022). Classifying Minimum Energy States for Interacting Particles: Regular Simplices. Communications in Mathematical Physics. Implies much of Theorem 2; Proposition 5 makes the overlap exact.",
      "url": "https://doi.org/10.1007/s00220-022-04564-x",
      "doi": "10.1007/s00220-022-04564-x"
    },
    {
      "citation": "Lim and McCann (2021). Isodiametry, Variance, and Regular Simplices from Particle Interactions. Archive for Rational Mechanics and Analysis. The t = ∞ edge case and the Jung-theorem connection.",
      "url": "https://doi.org/10.1007/s00205-021-01632-9",
      "doi": "10.1007/s00205-021-01632-9"
    },
    {
      "citation": "Björck (1956). Distributions of positive mass, which maximize a certain generalized energy integral. Arkiv för Matematik. Support theorems used in the proof of Theorem 2.",
      "url": "https://doi.org/10.1007/BF02589412",
      "doi": "10.1007/BF02589412"
    },
    {
      "citation": "Hille (1966). Some geometric extremal problems. Journal of the Australian Mathematical Society. The same Hölder step for configurations in a ball.",
      "url": "https://doi.org/10.1017/S1446788700004067",
      "doi": "10.1017/S1446788700004067"
    },
    {
      "citation": "Hille (1965). A note on transfinite diameters. Journal d'Analyse Mathématique. Possibly relevant; not read for the paper.",
      "url": "https://doi.org/10.1007/BF02806389",
      "doi": "10.1007/BF02806389"
    },
    {
      "citation": "Alexander and Stolarsky (1974). Extremal problems of distance geometry related to energy integrals. Transactions of the American Mathematical Society. Possibly relevant; not read for the paper.",
      "url": "https://doi.org/10.1090/S0002-9947-1974-0350629-3",
      "doi": "10.1090/S0002-9947-1974-0350629-3"
    },
    {
      "citation": "Jung (1901). Ueber die kleinste Kugel, die eine räumliche Figur einschliesst. Journal für die reine und angewandte Mathematik. Jung's theorem, recovered from Corollary 4 as t → ∞.",
      "url": "https://doi.org/10.1515/crll.1901.123.241",
      "doi": "10.1515/crll.1901.123.241"
    },
    {
      "citation": "Grove and Markvorsen (1995). New extremal problems for the Riemannian recognition program via Alexandrov geometry. Journal of the American Mathematical Society. Extent–excess argument sharpened by Theorems 11 and 13.",
      "url": "https://doi.org/10.1090/S0894-0347-1995-1276824-4",
      "doi": "10.1090/S0894-0347-1995-1276824-4"
    },
    {
      "citation": "Meyer (1981). Inequalities for the generalized transfinite diameter. Rocky Mountain Journal of Mathematics. Earlier inequalities for the same energies.",
      "url": "https://doi.org/10.1216/RMJ-1981-11-2-267",
      "doi": "10.1216/RMJ-1981-11-2-267"
    },
    {
      "citation": "Motzkin and Straus (1965). Maxima for Graphs and a New Proof of a Theorem of Turán. Canadian Journal of Mathematics. The quadratic-form maximum behind Lemma 3 and Remark 10.",
      "url": "https://doi.org/10.4153/CJM-1965-053-6",
      "doi": "10.4153/CJM-1965-053-6"
    },
    {
      "citation": "Leinster and Meckes (2016). Maximizing Diversity in Biology and Beyond. Entropy. A related maximum-diversity framework, mentioned as a pointer.",
      "url": "https://doi.org/10.3390/e18030088",
      "doi": "10.3390/e18030088"
    },
    {
      "citation": "Clark and Laugesen (2024). Maximizing Riesz capacity ratios: conjectures and theorems. arXiv:2410.14809 version 1, whose theorem and conjecture numbering the paper cites.",
      "url": "https://arxiv.org/abs/2410.14809v1",
      "doi": "10.48550/arXiv.2410.14809"
    },
    {
      "citation": "Fan (2026). Riesz capacity ratios with negative exponents. arXiv:2609.11186. Restates the two-point conjecture and compares a split tetrahedron with the ball.",
      "url": "https://arxiv.org/abs/2609.11186v1",
      "doi": "10.48550/arXiv.2609.11186"
    },
    {
      "citation": "ElonSG (2026). GitHub issue #22668, the-omega-institute/trureturing, opened 27 October 2026. Contains the same short proof of the two-point bound (Theorem 1).",
      "url": "https://github.com/the-omega-institute/trureturing/issues/22668",
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