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 $ X\subseteq\mathbb{R}^d$, we consider the functions

$\displaystyle \psi^{(2)}_X(u,v)=\lambda^d(X\cap (X-u)\cap (X-v)),\quad
\psi_X(u)=\lambda^d(X\cap (X-u)),$

where $ \lambda^d$ is the Lebesgue measure. These functions can be interpreted as volumes of erosions of $ X$ with three- or two-point test sets. The function $ \psi_X^{(2)}$ determines $ X$ uniquely up to translation and differences of measure zero [3]. We show that, if $ X$ belongs to the class $ {\cal U}_{PR}$ of locally finite unions of sets with positive reach and satisfies some further technical assumptions, then certain directional derivatives at 0 of $ \psi^{(2)}_X$ determine uniquely the surface area measure of $ X$. For the function $ \psi_X$ (known as set covariance of $ X$), we show that if $ X$ is a full-dimensional $ {\cal U}_{PR}$-set then the directional derivatives of $ \psi_X$ at 0 agree, up to a constant, with the total projection of $ X$ 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 $ \psi_X$ for the case of a smooth convex body $ X$ 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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