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Jin-ichi Itoh

Publications and source records attributed to Jin-ichi Itoh.

16 recordsLinked to original sources
arXiv2025

Hausdorff Stability of the Cut Locus Under $C^2$-Perturbations of the Metric

In this article, we prove the stability with respect to the Hausdorff metric $d_H$ of the cut locus $\mathrm{Cut}(p, \mathfrak{g})$ of a point $p$ in a compact Riemannian manifold $(M, \mathfrak{g})$ under $C^2$ perturbation of the metric. Specifically, given a sequence of metrics $\mathfrak{g}_i$ on $M$, converging to $\mathfrak{g}$ in the $C^2$ topology, and a sequence of points $p_i$ in $M$, converging to $p$, we show that $\lim_i d_{H}\left( \mathrm{Cut}(p_i, \mathfrak{g}_i), \mathrm{Cut}(p, \mathfrak{g}) \right) = 0$. Along the way, we also prove the continuous dependence of the cut time map on the metric.

math.DG↗
arXiv2021

Continuous Flattening of All Polyhedral Manifolds using Countably Infinite Creases

We prove that any finite polyhedral manifold in 3D can be continuously flattened into 2D while preserving intrinsic distances and avoiding crossings, answering a 19-year-old open problem, if we extend standard folding models to allow for countably infinite creases. The most general cases previously known to be continuously flattenable were convex polyhedra and semi-orthogonal polyhedra. For non-orientable manifolds, even the existence of an instantaneous flattening (flat folded state) is a new result. Our solution extends a method for flattening semi-orthogonal polyhedra: slice the polyhedron along parallel planes and flatten the polyhedral strips between consecutive planes. We adapt this approach to arbitrary nonconvex polyhedra by generalizing strip flattening to nonorthogonal corners and slicing along a countably infinite number of parallel planes, with slices densely approaching every vertex of the manifold. We also show that the area of the polyhedron that needs to support moving creases (which are necessary for closed polyhedra by the Bellows Theorem) can be made arbitrarily small.

cs.CG↗
arXiv2019

Locating diametral points

Let $K$ be a convex body in $\mathbb{R} ^d$, with $d = 2,3$. We determine sharp sufficient conditions for a set $E$ composed of $1$, $2$, or $3$ points of ${\rm bd}K$, to contain at least one endpoint of a diameter of $K$ (for $d=2,3$). We extend this also to convex surfaces, with their intrinsic metric. Our conditions are upper bounds on the sum of the complete angles at the points in $E$. We also show that such criteria do not exist for $n\geq 4$ points.

math.MG↗
arXiv2019

With respect to whom are you critical?

For any compact Riemannian surface $S$ and any point $y$ in $S$, $Q_y^{-1}$ denotes the set of all points in $S$, for which $y$ is a critical point. We proved \cite{BIVZ} together with Imre Bárány that card$Q_y^{-1} \geq 1$, and that equality for all $y\in S$ characterizes the surfaces homeomorphic to the sphere. Here we show, for any orientable surface $S$ and any point $y \in S$, the following two main results. There exist an open and dense set of Riemannian metrics $g$ on $S$ for which $y$ is critical with respect to an odd number of points in $S$, and this is sharp. Card$Q_y^{-1} \leq 5$ for the torus and card$Q_y^{-1} \leq 8g-5$ if the genus $g$ of $S$ is at least $2$. Properties involving points at globally maximal distance on $S$ are eventually presented.

math.GT↗
arXiv2019

The structure of the conjugate locus of a general point on ellipsoids and certain Liouville manifolds

It is well known since Jacobi that the geodesic flow of the ellipsoid is "completely integrable", which means that the geodesic orbits are described in a certain explicit way. However, it does not directly indicate that any global behavior of the geodesics becomes easy to see. In fact, it happened quite recently that a proof for the statement "The conjugate locus of a general point in two-dimensional ellipsoid has just four cusps" in Jacobi's Vorlesungen über dynamik appeared in the literature. In this paper, we consider Liouville manifolds, a certain class of Riemannian manifolds which contains ellipsoids. We solve the geodesic equations; investigate the behavior of the Jacobi fields, especially the positions of the zeros; and clarify the structure of the conjugate locus of a general point. In particular, we show that the singularities arising in the conjugate loci are only cuspidal edges and $D_4^+$ Lagrangian singularities, which would be the higher dimensional counterpart of Jacobi's statement.

math.DG↗
arXiv2014

Moderate smoothness of most Alexandrov surfaces

We show that, in the sense of Baire category, most Alexandrov surfaces with curvature bounded below by $κ$ have no conical points. We use this result to prove that at most points of such surfaces, the lower and the upper Gaussian curvatures are equal to $κ$ and $\infty$ respectively.

math.MG↗
arXiv2013

Every graph is a cut locus

We prove that every connected graph can be realized as the cut locus of some point on some Riemannian surface $S$ which, in some cases, has constant curvature. We study the stability of such realizations, and their generic behavior.

math.DG↗
arXiv2012

Source Unfoldings of Convex Polyhedra via Certain Closed Curves

We extend the notion of a source unfolding of a convex polyhedron P to be based on a closed polygonal curve Q in a particular class rather than based on a point. The class requires that Q "lives on a cone" to both sides; it includes simple, closed quasigeodesics. Cutting a particular subset of the cut locus of Q (in P) leads to a non-overlapping unfolding of the polyhedron. This gives a new general method to unfold the surface of any convex polyhedron to a simple, planar polygon.

cs.CG↗
arXiv2011

Orientable cut locus structures on graphs

We showed in another paper [arXiv:1103.1759] that every connected graph can be realized as the cut locus of some point on some riemannian surface $S$. Here, criteria for the orientability of $S$ are given, and are applied to classify the distinct, orientable, cut locus structures on graphs with four generating cycles.

math.DG↗
arXiv2011

Cut locus structures on graphs

Motivated by a fundamental geometrical object, the cut locus, we introduce and study a new combinatorial structure on graphs.

cs.DM↗
arXiv2011

On the number of cut locus structures on graphs

We proved in another paper that every connected graph can be realized as the cut locus of some point on some riemannian surface. Here we give upper bounds on the number of such realizations.

math.CO↗
arXiv2009

Star Unfolding Convex Polyhedra via Quasigeodesic Loops

We extend the notion of star unfolding to be based on a quasigeodesic loop Q rather than on a point. This gives a new general method to unfold the surface of any convex polyhedron P to a simple (non-overlapping), planar polygon: cut along one shortest path from each vertex of P to Q, and cut all but one segment of Q.

cs.CG↗
arXiv2009

The cut loci on ellipsoids and certain Liouville manifolds

We show that some riemannian manifolds diffeomorphic to the sphere have the property that the cut loci of general points are smoothly embedded closed disks of codimension one. Ellipsoids with distinct axes are typical examples of such manifolds.

math.DG↗
arXiv2008

Unfolding Convex Polyhedra via Quasigeodesic Star Unfoldings

We extend the notion of a star unfolding to be based on a simple quasigeodesic loop Q rather than on a point. This gives a new general method to unfold the surface of any convex polyhedron P to a simple, planar polygon: shortest paths from all vertices of P to Q are cut, and all but one segment of Q is cut.

cs.CG↗