Searcharxiv⌕ Search

arXiv subjects

Jun O'Hara

Publications and source records attributed to Jun O'Hara.

At least 19 recordsLinked to original sources
arXiv2026

Identification of generic polygonal domains by integral-geometric invariants

We study a reconstruction problem of planar domains from non-local integral-geometric invariants. We show that a generic polygonal domain, not necessarily convex, is uniquely determined, up to Euclidean isometry, by the interpoint distance distribution (IDD), which, for convex domains, is equivalent to the chord length distribution. Using a boundary representation of the Riesz energy function, we replace the IDD of the domain by an equivalent boundary IDD weighted by the scalar product of the outer unit normals, which we call $ν$-weighted IDD of the boundary. It enables us to reduce the problem to a one-dimensional problem. By analyzing jumps and blow-up terms of the second and third derivatives of the $ν$-weighted IDD of the boundary, we recover the side lengths, their cyclic incidence, and the exterior angles of the polygon. This can be considered as extending Waksman's classical generic reconstruction theorem for convex polygons to non-convex setting, as well as extending generic polygonal reconstruction from directional covariogram data to a one-dimensional invariant in which directional information has been integrated out.

math.MG↗
arXiv2026

Distinguishing finite metric spaces via similarity spectra

We study spectra and characteristic polynomials of similarity matrices associated with finite metric spaces, where the similarity matrix of a finite metric space $X=\{x_1,\dots,x_n\}$ is given by $\displaystyle Z(q)=(q^{d(x_i,x_j)})_{i,j}$. % We introduce two spectral invariants of finite metric spaces, the $q$-spectrum and the transition $q$-spectrum, defined respectively from $Z(q)$ and its transition matrix. In the case of graphs, these invariants recover the adjacency spectrum and the Laplacian spectrum in the limit $q\to0$. Our main result shows that the $q$-spectrum determines a large class of finite metric spaces under a natural nondegeneracy condition. We also prove that all four-point metric spaces are determined by their $q$-spectra. % The key observation is that the coefficients of the characteristic polynomial of $Z(q)$ encode cycle structures of the underlying metric space. % We further investigate the transition $q$-spectrum \jb{and show that strongly regular graphs with the same parameters have identical $q$-spectra and transition $q$-spectra, providing infinitely many non-isomorphic examples that cannot be distinguished by these invariants. Finally, we present computational examples comparing these invariants with classical graph spectra.}

math.MG↗
arXiv2026

Distinguishing regular polygons, cycle graphs, and circular metric spaces by the distance multiset and magnitude

We investigate how effectively finite metric spaces can be distinguished by distance-based invariants. As model spaces, we consider regular polygons, cycle graphs, and their generalization, circular metric spaces, and as invariants we consider the distance multiset, magnitude, and magnitude homology. We construct explicit families of homometric but non-congruent circular metric spaces, and in many even cases these examples also have the same magnitude as the original space. We prove that regular polygons are determined by the distance multiset among planar metric spaces, but not in general. We also determine, for several values of $n$, whether regular $n$-gons and $n$-cycle graphs are determined by magnitude.

math.MG↗
arXiv2026

CR-invariant energy of Legendrian knots in the Heisenberg group

We introduce an energy functional for Legendrian knots in the 3-dimensional Heisenberg group $\mathcal{H}$, which serves as a sub-Riemannian analog of the Möbius invariant knot energy in Euclidean 3-space introduced by the second author. The energy is obtained by regularizing a divergent integral of the potential of order -2 with respect to the Korányi distance on $\mathcal{H}$; this choice of distance is essential for the energy to be invariant under the action of PU(2,1). We characterize $\mathbb{R}$-circles in $\mathcal{H}$ as the minimizers of the energy, and establish a Heisenberg analog of the Doyle--Schramm cosine formula. We also show that the energy integrand admits an expression in terms of a complex-valued 2-form on the complement of the diagonal in $\mathcal{H}\times\mathcal{H}$, providing a partial analog of the infinitesimal cross ratio interpretation known from the classical setting.

math.GT↗
arXiv2025

Magnitude function determines generic finite metric spaces

We give sufficient conditions for a finite metric space to be determined by the magnitude function. In particular, a generic finite metric space such that the distances between the points are rationally independent is determined by the asymptotic behavior of the magnitude function.

math.MG↗
arXiv2022

Uniqueness of centers of nearly spherical bodies

An $r^{\an}$-center of a compact body $\Om$ in an $n$ dimensional Euclidean space is a point that gives an extremal value of the regularized Riesz potential, which is the (Hadamard regularization of) integration on $\Om$ of the distance from the point to the power $\an$. We show that for any real number $\la$ if a compact body is sufficiently close to a ball in the sense of asphericity then the $r^{\an}$-center is unique. We also study the regularized potentials of a unit ball.

math.DG↗
arXiv2022

Residues of manifolds

The Riesz $z$-energy of a manifold $X$ is the integration of the distance between two points to the power $z$ over the product space $X\times X$. Considered as a function of a complex variable $z$, it can be generalized to a meromorphic function by analytic continuation, which we will call the meromorphic energy function of $X$. It has only simple poles at some negative integers. The residues of a manifold $X$ are the residues of the meromorphic energy function. For example, the volume and the Willmore energy for surfaces in $\mathbb{R}^3$ can be obtained as residues. In this paper we first show the Möbius invariance of the residue at $z=-2\dim X$ of a closed submanifold or a compact body in a Euclidean space. We introduce the relative residues for compact bodies and weighted residues, and show that the scalar curvature and the mean curvature as well as the Euler characteristic of compact bodies of dimension less than $4$ can be expressed in terms of residues and local residues. We study the order of differentiaion of a local defining function of $X$ that is necessary to obtain the (global) residues when $X$ is a closed submanifold of a Euclidean space. We also show the inclusion-exclusion principle. Residues appear to be similar to quantities obtained by asymptotic expansion such as intrinsic volumes (Lipschitz-Killing curvatures), spectra of Laplacian, and the Graham-Witten energy. We show that residues are independent from them. Finally we introduce a \M invariant principal curvature energy for $4$-dimensional hypersurfaces in $\mathbb{R}^5$, and express the Graham-Witten energy in terms of the residues, Weyl tensor, and this Möbius invariant principal curvature energy.

math.DG↗
arXiv2021

Strict power concavity of a convolution

We give a sufficient condition for the strict parabolic power concavity of the convolution in space variable of a function defined on $\mathbb{R}^n \times (0,+\infty)$ and a function defined on $\mathbb{R}^n$. Since the strict parabolic power concavity of a function defined on $\mathbb{R}^n \times (0,+\infty)$ naturally implies the strict power concavity of a function defined on $\mathbb{R}^n$, our sufficient condition implies the strict power concavity of the convolution of two functions defined on $\mathbb{R}^n$. As applications, we show the strict parabolic power concavity and strict power concavity in space variable of the Gauss--Weierstass integral and the Poisson integral for the upper half-space.

math.AP↗
arXiv2021

Self-repulsiveness of energies for closed submanifolds

We show that the regularized Riesz $\a$-energy for closed submanifolds $M$ in $\RR^n$ blows up as $M$ degenerates to have double points if $\a\le-2\dim M$. This gives theoretical foundation of numerical experiments to evolve surfaces to decrease the energy which have been carried out since 90's.

math.GT↗
arXiv2020

Pompeiu's theorem and the moduli space of triangles

We introduce a kind of converse of Pompeiu's theorem. Fix an equilateral triangle $\triangle A_0B_0C_0$, then for any triangle $\triangle ABC$ there is a unique point $P$ inside the circumcircle $Γ_0$ of $\triangle A_0B_0C_0$ such that a triangle with edge lengths $PA_0, PB_0$, and $PC_0$ is similar to $\triangle ABC$. It follows that an open disc inside $Γ_0$ can be considered as a moduli space of similarity classes of triangles. We show that it is essentially equivalent to another moduli space based on a shape function of triangles which has been used in preceding studies.

math.HO↗
arXiv2020

Möbius invariant metrics on the space of knots

We give a condition for a function to produce a Möbius invariant weighted inner product on the tangent space of the space of knots, and show that some kind of Möbius invariant knot energies can produce Möbius invariant and parametrization invariant weighted inner products. They would give a natural way to study the evolution of knots in the framework of Möbius geometry.

math.DG↗
arXiv2020

Regularized Riesz energies of submanifolds

Given a closed submanifold, or a compact regular domain, in euclidean space, we consider the Riesz energy defined as the double integral of some power of the distance between pairs of points. When this integral diverges, we compare two different regularization techniques (Hadamard's finite part and analytic continuation), and show that they give essentially the same result. We prove that some of these energies are invariant under Moebius transformations, thus giving a generalization to higher dimensions of the Moebius energy of knots.

math.DG↗
arXiv2018

Regularization of self inductance

We introduce several methods to define the self-inductance of a single loop as the regularization of divergent integrals which we obtain by applying Neumann (or Weber) formula for the mutual inductance of a pair of loops to the case when two loops are identical.

math.DG↗
arXiv2017

Characterization of balls by generalized Riesz energy

We show that balls, circles and 2-spheres can be identified by generalized Riesz energy among compact submanifolds of the Euclidean space that are either closed or with codimension 0, where the Riesz energy is defined as the double integral of some power of the distance between pairs of points. As a consequence, we obtain the identification by the interpoint distance distribution.

math.DG↗
arXiv2016

Equisectional equivalence of triangles

We study equivalence relation of the set of triangles generated by similarity and operation on a triangle to get a new one by joining division points of three edges with the same ratio. Using the moduli space of similarity classes of triangles introduced by Nakamura and Oguiso, we give characterization of equivalent triangles in terms of circles of Apollonius (or hyperbolic pencil of circles) and properties of special equivalent triangles. We also study rationality problem and constructibility problem.

math.MG↗
arXiv2016

Renormalization of potentials and generalized centers

We generalize the Riesz potential of a compact domain in $\mathbb{R}^{m}$ by introducing a renormalization of the $r^{α-m}$-potential for $α\le0$. This can be considered as generalization of the dual mixed volumes of convex bodies as introduced by Lutwak. We then study the points where the extreme values of the (renormalized) potentials are attained. These points can be considered as a generalization of the center of mass. We also show that only balls give extreme values among bodied with the same volume.

math.DG↗
arXiv2013

Energy of tori of revolution

We show that the surface energy introduced by Auckly and Sadun attains the minimum value at the Clifford torus among tori of revolution.

math.DG↗