On Projection Bodies of Order One

http://dx.doi.org/10.4153/CMB-2009-038-6
Canad. Math. Bull. 52(2009), 349-360

  Published:2009-09-01
 Printed: Sep 2009

The projection body of order one $\Pi_1K$ of a convex body $K$ in $\R^n$ 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 $\Pi_1K$, 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 $n-1$.