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On Projection Bodies of Order One

Open Access article
 Printed: Sep 2009
  • Stefano Campi
  • Paolo Gronchi
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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 $n-1$.
MSC Classifications: 52A40 show english descriptions Inequalities and extremum problems 52A40 - Inequalities and extremum problems

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