Two Dimensional Zonoids and Chebyshev Measures

✍ Scribed by Stefano Bianchini; Raphaël Cerf; Carlo Mariconda

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 231 KB
• Volume: 211
• Category: Article
• ISSN: 0022-247X
• DOI: 10.1006/jmaa.1997.5486

✦ Synopsis

We give an alternative proof to the well known fact that each convex compact centrally symmetric subset of ‫ޒ‬ 2 containing the origin is a zonoid, i.e., the range of a two dimensional vector measure, and we prove that a two dimensional zonoid whose boundary contains the origin is strictly convex if and only if it is the range of a Chebyshev measure. We give a condition under which a two dimensional vector measure admits a decomposition as the difference of two Chebyshev measures, a necessary condition on the density function for the strict convexity of the range of a measure and a characterization of two dimensional Chebyshev measures.