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Estimation of oriented direction distribution of a planar body. (English) Zbl 0861.60023

For a planar set \(X\) from the convex ring \(\mathcal R\), the estimation of the oriented direction distribution is discussed. The latter is the normalized version of the additively extended surface area measure \(\sigma(X,\cdot)\) of \(X\). The proposed method proceeds in two steps and rests on the connection between surface area measures and mixed areas. In the first step, a comparison between the area \(A(X+\varepsilon K)\) of the outer parallel set \(X+\varepsilon K\), \(K\) a planar convex body, \(\varepsilon>0\), with the Steiner formula on \(\mathcal R\) is used to show that \({1\over 2\varepsilon}(A(X+\varepsilon K)-A(X))\) tends to the mixed area \(A(X,K)\), as \(\varepsilon\to 0\). In the second step, suitable test sets \(K\) are used to invert the classical integral formula, which relates \(\sigma(X,\cdot)\) with \(A(X,K)\). The author also gives bounds for the estimation bias and the estimation variance, if point counting methods are applied.
Reviewer: W.Weil (Karlsruhe)

MSC:

60D05 Geometric probability and stochastic geometry
52A39 Mixed volumes and related topics in convex geometry
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