Geometric properties of a sine function extendable to arbitrary normed planes

Abstract

Abstract. In this paper we study a metric generalization of the sine function which can be extended to arbitrary normed planes. We derive its main properties and give also some characterizations of Radon planes. Furthermore, we prove that the existence of an angular measure which is "well-behaving" with respect to the sine is only possible in the Euclidean plane, and we also define some new constants that estimate how non-Radon or non-Euclidean a normed plane can be. Sine preserving self-mappings are studied, and a complete description of the linear ones is given. In the last section we exhibit a version of the Law of Sines for Radon planes.

Angles in normed spaces

“…Proof. It is clear that ang s is continuous for any pair x, y ∈ X o such that [x, b(y)] = 0 (this follows from the continuity of the sine function proved in [7]). Hence, we just have to consider a pair x, y with x ⊣ B y and prove that ang s is continuous in both entries.…”

“…Proof. It is known (see [7]) that the sine function is symmetric only in Radon planes. The result follows.…”

“…The result follows. Still assuming that x, y ∈ S are two unit vectors such that x ⊣ B y, recall that the Busemann angular bisector of the angle < ) xoy is the ray with direction x + y, and hence Proposition 3.8 in [7] states that s(x, x + y) = s(y, x + y). If ang s is additive, we have ang s (x, x + y) + ang s (x + y, y) = ang s (x, y) = π 2 .…”

A new geometric view on Sturm–Liouville eigenvalue problems

“…We prove in Section 4 that if the norm is Radon, then the cycloids associated to the eigenvalue λ = 1 are given by the trigonometric functions for normed planes studied in [5] and [6]. This is similar to what happens when the geometry is Euclidean, and it is not true if the norm is not Radon.…”

“…Geometrically, the Minkowskian sine function sm(x, y) is the (signed) distance in the norm from the origin to the line R ∋ t → x + ty. For the proof, and also for related discussions involving the Minkowskian sine function, we refer to [5]. Notice that the Minkowskian sine function can be defined also for normed planes which are not smooth or strictly convex.…”