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SUCCESSIVE RADII OF FAMILIES OF CONVEX BODIES

Part of: General convexity

Published online by Cambridge University Press:  15 December 2014

BERNARDO GONZÁLEZ ,
MARÍA A. HERNÁNDEZ CIFRE  and
AICKE HINRICHS
BERNARDO GONZÁLEZ
Affiliation:
Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100-Murcia, Spain email bgmerino@gmail.com
MARÍA A. HERNÁNDEZ CIFRE*
Affiliation:
Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Murcia, Spain email mhcifre@um.es
AICKE HINRICHS
Affiliation:
Institut für Analysis, Johannes Kepler Universität Linz, Altenberger Str. 69, 4040 Linz, Austria email aicke.hinrichs@jku.at
*
✉mhcifre@um.es
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Abstract

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We study properties of the so-called inner and outer successive radii of special families of convex bodies. First we consider the balls of the $p$-norms, for which we show that the precise value of the outer (inner) radii when $p\geq 2$ ($1\leq p\leq 2$), as well as bounds in the contrary case $1\leq p\leq 2$ ($p\geq 2$), can be obtained as consequences of known results on Gelfand and Kolmogorov numbers of identity operators between finite-dimensional normed spaces. We also prove properties that successive radii satisfy when we restrict to the families of the constant width sets and the $p$-tangential bodies.

Keywords

successive inner and outer radiiGelfand and Kolmogorov numbersp-ballsconstant width setstangential bodies

MSC classification

Primary: 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces)
Secondary: 52A40: Inequalities and extremum problems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 91 , Issue 2 , April 2015 , pp. 331 - 344
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

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