A note on Reeb spaces of some explicit real analytic functions

Naoki Kitazawa

Home/Archive/arXiv

arXiv · 2601.11648

A note on Reeb spaces of some explicit real analytic functions

Naoki Kitazawa

View original source ↗Read source document ↗Export metadata ↓

Abstract

Reeb spaces of smooth functions are fundamental and strong tools in understanding manifolds via smooth functions with mild critical points. They are defined as the natural spaces of all connected components of level sets. They are also important objects in related studies. Realization of graphs as Reeb spaces of smooth functions of certain nice classes is of such studies. In this paper, we present Reeb spaces of explicit real analytic functions which are not finite graphs. Related problems were started by Sharko, in 2006, who has studied smooth functions with critical points represented by certain elementary polynomials, and followed by a study of Masumoto and Saeki, which is on smooth functions on closed surfaces under an extended situation, and a study of Michalak, which is on Morse functions on closed manifolds. The author has contributed to this by respecting topologies of level sets, and real algebraic construction.

Explore related subjects

math.GM

Keep this discovery

BibTeX RIS

Naoki Kitazawa. 2026-01-15. A note on Reeb spaces of some explicit real analytic functions. https://arxiv.org/abs/2601.11648

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Average Chord Lengths in a Triangle

Let $P$ be a point inside a triangle $T$. We consider the average length of the chords of $T$ through $P$, where the direction of the chord is chosen uniformly. An elementary formula is obtained in terms of the distances from $P$ to the sides and vertices of the triangle. Several classical triangle centers give especially simple specializations. For example, if $I$ is the incenter, then \[ M_T(I)=\frac{2r}{\pi} \log\left(\cot\frac A4\cot\frac B4\cot\frac C4\right). \] Our main result is the sharp inequality \[ M_T(P)\le \frac{p}{\pi\sqrt3}\log(2+\sqrt3), \] valid simultaneously for every triangle of perimeter $p$ and every interior point $P$. Thus, among all such pairs $(T,P)$, the largest possible average chord length occurs only when $T$ is equilateral and $P$ is its center. The proof is an elementary symmetrization argument. We close with brief remarks relating the problem to the radial center of a convex body, the electrostatic potential center of a triangle, and dual quermassintegrals.

math.GM↗

arXiv2026

The decrease in sectional curvature of the Fisher--Rao manifold of beta distributions

In the paper, the author develops a general convexity lemma and applies it to demonstrate the monotonic decrease of a function involving polygamma functions, arising from the sectional curvature of the Fisher--Rao manifold of beta distributions.

math.GM↗

arXiv2026

A Proof of Liu's Conjecture on the Fundamental Triangle Inequality

Let $a,b,c$ be the side lengths of a triangle, and let $R$ and $r$ denote its circumradius and inradius, respectively. We prove a conjecture of Liu stating that \[\sum_{\mathrm{cyc}} \left(\frac{a(b+c-a)}{bc}\right)^k \geq 2+\left(\frac{2r}{R}\right)^k,~~k>1, \] with the reverse inequality for $0<k<1$. The proof reduces the problem to three positive variables with fixed sum and product. We also determine the equality cases.

math.GM↗