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We study the problem whether regular polygons and cycle graphs can be identified by the magnitude or the discrete Riesz energy. We give a positive answer in some cases and a negative answer in other cases by giving a way to construxt non-isometric spaces with the same magnitude or the discrete Riesz energy., Comment: 16 pages, 16 figures

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., Comment: 20 pages, 4 figures

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., Comment: 19 pages, 4 figures

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 M\"obius 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 M\"obius invariant principal curvature energy., Comment: 51 pages

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., Comment: 19 pages

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., Comment: 14 pages, two figures

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 $\Gamma_0$ 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 $\Gamma_0$ 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., Comment: 10 pages, 7 figures

We give a condition for a function to produce a M\"obius invariant weighted inner product on the tangent space of the space of knots, and show that some kind of M\"obius invariant knot energies can produce M\"obius invariant and parametrization invariant weighted inner products. They would give a natural way to study the evolution of knots in the framework of M\"obius geometry., Comment: 14 pages

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., Comment: 14pages

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., Comment: 9 pages, 2 figures

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., Comment: 10 pages, 6 figures

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., Comment: 21 pages. The proof of Theorem 4.10 in the previous version is corrected (now Theorem 4.16 in the present version). The previous version appeared in Math. Nachr. 291 (2018), 1356-1373, doi: 10.1002/mana.201600083

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

The {\em minimal unfolded region} (or the {\em heart}) of a bounded subset $\Om$ in the Euclidean space is a subset of the convex hull of $\Om$ the definition of which is based on reflections in hyperplanes. It was introduced to restrict the location of the points that give extreme values of certain functions, such as potentials whose kernels are monotone functions of the distance, and solutions of differential equations to which Aleksandrov's reflection principle can be applied. %the temperature of a heat conductor with some initial-boundary condition, in which case the points are called the hot spots. We show that the minimal unfolded regions of the convex hull and parallel bodies of $\Om$ are both included in that of $\Om$., Comment: to appear in Hokkaido Math. J

We show the uniqueness of the radial centers of any order $\alpha$ of a parallel body of a convex body $\Omega$ in $\mathbb R^m$ at distance $\delta$ if $\delta$ is greater than the diameter of $\Omega$ multiplied by a constant which depends only on the dimension $m$., Comment: 13 pages, 8 figures

We study the configuration space of equilateral and equiangular spatial hexagons for any bond angle by giving explicit expressions of all the possible shapes. We show that the chair configuration is isolated, whereas the boat configuration allows one-dimensional deformations which form a circle in the configuration space., Comment: 10 pages, 15 figures

We study optimal double helices with straight axes (or the fattest tubes around them) computationally using three kinds of functionals; ideal ones using ropelength, best volume packing ones, and energy minimizers using two one-parameter families of interaction energies between two strands of types $r^{-\alpha}$ and $\frac1r\exp(-kr)$. We compare the numerical results with experimental data of DNA., Comment: 10 pages, 16 figures, to appear in Progress of Theoretical Physics Supplement. Minor change and reference updated

We show that the the image of the regular projection of a vertex of a cone over a triangle that minimizes the ratio of the cube of the area of the boundary of the cone and the square of the volume of the cone coincides with the incenter., Comment: 3 pages

We define and study a M\"obius invariant energy associated to planar domains, as well its generalization to space curves. This generalization is a M\"obius version of Banchoff-Pohl's notion of area enclosed by a space curve. A relation with Gauss-Bonnet theorems for complete surfaces in hyperbolic space is also described., Comment: 36 pages Minor changes in version 2

We generalize the Riesz potential of a compact domain in $\mathbb{R}^{m}$ by introducing a renormalization of the $r^{\alpha-m}$-potential for $\alpha\le0$. 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., Comment: Adv. Appl. Math. 48 (2012), 365--392 Figure 11 has been corrected after publication. Theorem 3.12 and the exposition of Lemma 2.15 are modified in version 8

The set of osculating circles of a given curve in $\SS^3$ forms a curve in the set of oriented circles in $\SS^3$. We show that its "${\frac12}$-dimensional measure" with respect to the pseudo-Riemannian structure of the set of circles is proportional to the conformal arc-length of the original curve, which is a conformally invariant local quantity discovered in the first half of the last century., Comment: 39 pages, 11 figures. To appear in Comm. Math. Helv

A two-component link produces a torus as the product of the component knots in a two-point configuration space of a three-sphere. This space can be identified with a cotangent bundle and also with an indefinite Grassmannian. We show that the integration of the absolute value of the canonical symplectic form is equal to the area of the torus with respect to the pseudo-Riemannian structure, and that it attains the minimum only at the "best" Hopf links., Comment: to appear in Tohoku Math. J

Let $\{P_1, P_2, P_3, P_4\}$ be a quadruplet of points in $S^3$ . We define a ``dual'' quadruplet of it in a conformal geometric way. We show that the dual of a dual quadruplet coincides with the original one. We also show that the cross ratio of the dual quadruplet is equal to the complex conjugate of that of the original one., Comment: This is a second version. A very short section was added at the end, where a cocircular case, which is a special case, was studied. A result in elementary geometry, especially, inversive geometry. 9 pages, 9 figures

This is a survey article on two topics. The Energy E of knots can be obtained by generalizing an electrostatic energy of charged knots in order to produce optimal knots. It turns out to be invariant under Moebius transformations. We show that it can be expressed in terms of the infinitesimal cross ratio, which is a conformal invariant of a pair of 1-jets, and give two kinds of interpretations of the real part of the infinitesimal cross ratio., Comment: This is the version published by Geometry & Topology Monographs on 19 March 2008

We study a 1-form which can be given by a vector in a conformally invariant way. We then study conformally invariant functionals associated to a ``Y-diagram'' on the space of knots which are made from the 1-form., Comment: 11 pages, 4 fiugres

This article serves as errata of the book "Energy of knots and conformal geometry", Series on Knots and Everything Vol. 33, World Scientific, Singapore, 304 pages, (2003). (ver. 27/05/2007), Comment: 11 pages, 2 figures Updated Sept. 2018. Added an idea of the proof of self-repulsiveness of surface energies

The configuration space of the mechanism of a planar robot is studied. We consider a robot which has $n$ arms such that each arm is of length 1+1 and has a rotational joint in the middle, and that the endpoint of the $k$-th arm is fixed to $Re^{\frac{2(k-1)\pi}ni}$. Generically, the configuration space is diffeomorphic to an orientable closed surface. Its genus is given by a topological way and a Morse theoretical way. The homeomorphism types of it when it is singular is also given., Comment: 33 pages, 41 figures