How the elliptic integrals \(K\) and \(E\) arise from circles and points in the Minkowski plane
type: Item
instance of: scholarly article
title: How the elliptic integrals \(K\) and \(E\) arise from circles and points in the Minkowski plane
zbMATH Open document ID: 0808.52004
DOI: 10.1007/BF01222663
author: Mostafa Ghandehari
author: Dave Logothetti
published in: Journal of Geometry
publication date: 1995-03-16T00:00:00Z
review text: Let \(K\) be a centrally symmetric plane convex body, centered at the origin. Thinking \(K\) as the unit disk for a 2-dimensional Banach space (Minkowski plane), let \(\sigma(K)\) denote the length of \(K\) computed using the metric induced by \(K\). Then \(\sigma(K)\) is called the self- circumference of \(K\). If \(K\) is not necessarily centrally symmetric and \(z\) is any interior point of \(K\), then \(\sigma_ +(K,z)\) and \(\sigma_ -(K,z)\) are defined as the respective positive and negative self- circumferences of \(K\) at \(z\). It is shown that if \(K = B = \) Euclidean unit circle and \(z\) is any interior point of \(B\), \(\sigma_ +(B, z)\) can be expressed in terms of the complete elliptic integral of the second kind. Next the author defines a function \(\tau (B, z)\) that is given by a complete elliptic integral of the first kind. If \(L(K)\) denotes the euclidean length of \(K\) and \(L^*(K)\) that of the polar dual \(K^*\) of \(K\), then the inequality \(L(K)L^*(K) \geq 4 \pi^ 2\) is used. Asymptotic results as \(z\) approaches the boundary of \(B\) are included. An affine transformation is used to discuss similar results for an ellipse.
Mathematics Subject Classification ID: 52A21
Mathematics Subject Classification ID: 53A15
zbMATH DE Number: 621989
zbMATH Keywords: elliptic integrals
zbMATH Keywords: affine invariants
zbMATH Keywords: Minkowski plane
zbMATH Keywords: self- circumference
reviewed by: L. A. Santaló
MaRDI profile type: Publication
cites work: Gauss, Landen, Ramanujan, the Arithmetic-Geometric Mean, Ellipses, π, and the Ladies Diary
cites work: Some properties of the self-circumference of convex sets
cites work: Q5824493
cites work: Steinhardt's inequality in the Minkowski plane
cites work: Q4401917
full work available at URL: https://doi.org/10.1007/bf01222663
OpenAlex ID: W2051068839
description: scientific article; zbMATH DE number 621989