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On Projection Bodies of Order One

Published online by Cambridge University Press:  20 November 2018

Stefano Campi  and
Paolo Gronchi
Stefano Campi
Affiliation:
Dipartimento di Ingegneria dell’Informazione, Università degli Studi di Siena, Via Roma 56, 53100 Siena, Italy e-mail: campi@dii.unisi.it
Paolo Gronchi
Affiliation:
Dipartimento di Matematica e Applicazioni per l’Architettura, Università degli Studi di Firenze, Piazza Ghiberti 27, 50122 Firenze, Italy e-mail: paolo@fi.iac.cnr.it
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Abstract

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The projection body of order one ${{\Pi }_{1}}K$ of a convex body $K$ in ${{\mathbb{R}}^{n}}$ is the body whose support function is, up to a constant, the average mean width of the orthogonal projections of $K$ onto hyperplanes through the origin.

The paper contains an inequality for the support function of ${{\Pi }_{1}}K$, which implies in particular that such a function is strictly convex, unless $K$ has dimension one or two. Furthermore, an existence problem related to the reconstruction of a convex body is discussed to highlight the different behavior of the area measures of order one and of order $n\,-\,1$.

Keywords

52A40
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 52 , Issue 3 , 01 September 2009 , pp. 349 - 360
Copyright
Copyright © Canadian Mathematical Society 2009

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