bibkey: bilykferizovicglazyrinmatzkeparkvlasiuk2022multivariate
authors: D. Bilyk, D. Ferizović, A. Glazyrin, R. Matzke, J. Park, O. Vlasiuk
year: 2022
title: Potential theory with multivariate kernels
doi: 10.1007/s00209-022-03000-z
url: https://arxiv.org/abs/2104.03410
claim: Normalized surface measure minimizes a continuous symmetric rotationally invariant multivariate energy if and only if its averaged pair kernel is conditionally positive definite.
strata_touched:
  - D5/S3/Analytic/MultivariateEnergy/Defs
  - D5/S3/Analytic/MultivariateEnergy/Moments
  - D5/S3/Analytic/MultivariateEnergy/Energy
  - D5/S3/Analytic/MultivariateEnergy/Result
license: citation-only
triage: anchor

Potential theory with multivariate kernels

D. Bilyk, D. Ferizović, A. Glazyrin, R. Matzke, J. Park and O. Vlasiuk, Potential theory with multivariate kernels, Math. Z. 301 (2022).

Definitions 2.1–2.2 specify conditional positive definiteness using finite signed Borel measures and the corresponding energy minimization condition. Equation (5.1) specifies rotational invariance. Section 5, paragraph before Theorem 5.3, states:

We suspect that the property that σ minimizes I_K is equivalent to the fact that U_K^{σ^{n−2}} is conditionally positive definite, i.e. the two-input energy I_{U_K^{σ^{n−2}}} is minimized by σ. This conjecture is supported by all the examples known to us.

The source establishes necessity and a local converse in Theorem 5.3. The global converse is refuted by the centered matrix family in the cited modules: for three inputs, dimension at least three, and 0 < β < (d−2)/d, the averaged kernel is conditionally positive definite while a Dirac probability measure has strictly smaller energy than normalized surface measure.

Verified locator

• DOI: https://doi.org/10.1007/s00209-022-03000-z
• URL: https://arxiv.org/abs/2104.03410
• Location: Section 5, paragraph before Theorem 5.3; Definitions 2.1–2.2; (5.1).