## 15. ÖMG-Kongress Jahrestagung der Deutschen Mathematikervereinigung

#### 16. bis 22. September 2001 in Wien

Sektion 5 - Geometrie
Montag, 17. September 2001, 16.00, Hörsaal 50

Set covariance and finite test sets

Jan Rataj, Charles University, Prag

Given a nonempty compact subset , we consider the functions

where is the Lebesgue measure. These functions can be interpreted as volumes of erosions of with three- or two-point test sets. The function determines uniquely up to translation and differences of measure zero [3]. We show that, if belongs to the class of locally finite unions of sets with positive reach and satisfies some further technical assumptions, then certain directional derivatives at 0 of determine uniquely the surface area measure of . For the function (known as set covariance of ), we show that if is a full-dimensional -set then the directional derivatives of at 0 agree, up to a constant, with the total projection of in the given direction (this fact was known at least for the case of convex bodies). Further, we calculate certain second and third order directional derivatives of for the case of a smooth convex body with positive Gauss curvature. This yields a partial answer (in the planar smooth case) to the problem whether the set covariance determines a convex body up to translation and central reflection (see [1,2]).

 [1] A.J. Cabo, A.J. Baddeley: Line transects, covariance functions and set convergence. Adv. Appl. Probab. 27 (1995), 585-605 [2] W. Nagel: Orientation-dependent chord length distributions characterize convex polygons. J. Appl. Prob. 30 (1993), 730-736 [3] J. Rataj: Characterization of compact sets by their dilation volume. Math. Nachr. 173 (1995), 287-295

 E-Mail: rataj@karlin.mff.cuni.cz Homepage: www.karlin.mff.cuni.cz/~rataj

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