On Projection Bodies of Order One
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 $n-1$.
52A40 - Inequalities and extremum problems