Remarks on retracting balls on spherical caps in \(c_{0}\), \(c\), \(l^{\infty }\) spaces

Kazimierz Goebel

doi:10.17951/a.2020.74.1.45-55

Abstract

For any infinite dimensional Banach space there exists a lipschitzian retraction of the closed unit ball B onto the unit sphere S. Lipschitz constants for such retractions are, in general, only roughly estimated. The paper is illustrative. It contains remarks, illustrations and estimates concerning optimal retractions onto spherical caps for sequence spaces with the uniform norm.

Highlights

Let X be a Banach space with the norm ·, the closed unit ball B and unit sphere S
Dim X = ∞ implies the existence of a retraction R : B → S satisfying on B the Lipschitz condition
As declared at the beginning, we shall modify the notion of spherical caps and try to estimate the optimal Lipschitz constant of corresponding retractions

Summary

Introduction

Let X be a Banach space with the norm · , the closed unit ball B and unit sphere S. Retraction, Lipschitz constant, radial projection, truncation, spherical cap. For a Hilbert space it is Combining the radial projection with the identity on the ball B, we get the retraction P : X → B:. Fact 2: In the spaces under our consideration there is another natural retraction T : X → B Fact 3: In our setting the radial projection P maps Dr onto the sphere S having the Lipschitz constant k(P ) =. As declared at the beginning, we shall modify the notion of spherical caps and try to estimate the optimal Lipschitz constant of corresponding retractions. Since our spaces have big flat spots on the bottom and the top of the sphere, we shall consider three types of spherical caps.

Boxes of various height

Closing the sphere caps