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On Projection Bodies of Order One

  • Stefano Campi (a1) and Paolo Gronchi (a2)

Abstract

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 $n\,-\,1$ .

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References

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On Projection Bodies of Order One

  • Stefano Campi (a1) and Paolo Gronchi (a2)

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