Metrics in the family of star bodies
Abstract
In this paper we consider different ways of introducing metrics in the family of star bodies. We begin with basic properties of metrics commonly used. Then we use selectors (see Definition 4.1) to extend the radial metric (see Definition 3.2) over the class of all star bodies in n-dimensional euclidean space. This way we obtain two slightly different metrics δφ and δφker (see Definition 4.2). We show that both metric spaces obtained are separable. Further, we present an analytic approach to the radial function. It enables us to define the metric δφL ' (see Definition 5.3). We prove that if we use δφL ' in the family of star bodies whose kernels have non-empty interiors, then the mapping K 7→ kerK is continuous.
© 2013 by Walter de Gruyter GmbH & Co.
Articles in the same Issue
- Masthead
- A class of lattices and boolean functions related to the Manickam–Miklös–Singhi conjecture
- Blocking semiovals containing conics
- On universal covers in normed planes
- Stability results for some classical convexity operations
- Pseudo-embeddings and pseudo-hyperplanes
- Asymptotic estimates on the time derivative of entropy on a Riemannian manifold
- Metrics in the family of star bodies
- Ellipsoid characterization theorems
- Logarithmic limit sets of real semi-algebraic sets
Articles in the same Issue
- Masthead
- A class of lattices and boolean functions related to the Manickam–Miklös–Singhi conjecture
- Blocking semiovals containing conics
- On universal covers in normed planes
- Stability results for some classical convexity operations
- Pseudo-embeddings and pseudo-hyperplanes
- Asymptotic estimates on the time derivative of entropy on a Riemannian manifold
- Metrics in the family of star bodies
- Ellipsoid characterization theorems
- Logarithmic limit sets of real semi-algebraic sets
