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Keywords:
convex body; set with positive reach; normal measure; set covariance
convex body; set with positive reach; normal measure; set covariance
Summary:
The information contained in the measure of all shifts of two or three given points contained in an observed compact subset of $\mathbb{R}^d $ is studied. In particular, the connection of the first order directional derivatives of the described characteristic with the oriented and the unoriented normal measure of a set representable as a finite union of sets with positive reach is established. For smooth convex bodies with positive curvatures, the second and the third order directional derivatives of the characteristic is computed.
References:
[1] G. Bianchi, F. Segala, and A. Volčič: The solution of the covariogram problem for plane $C^2_+$ convex bodies. J. Differential Geom. 60 (2002), 177–198. DOI 10.4310/jdg/1090351101 | MR 1938112
[2] H. Federer: Curvature measures. Trans. Amer. Math. Soc. 93 (1959), 418–491. DOI 10.1090/S0002-9947-1959-0110078-1 | MR 0110078 | Zbl 0089.38402
[3] H. Federer: Geometric Measure Theory. Springer-Verlag, Berlin, 1969. MR 0257325 | Zbl 0176.00801
[4] A. Lešanovský and J. Rataj: Determination of compact sets in Euclidean spaces by the volume of their dilation. DIANA III (Proc. conf.), MÚ ČSAV, Praha, 1990, pp. 165–177.
[5] A. Lešanovský, J. Rataj and S. Hojek: 0-1 sequences having the same numbers of (1-1) couples of given distances. Math. Bohem. 117 (1992), 271–282. MR 1184540
[6] G. Matheron: Random Sets and Integral Geometry. J. Wiley, New York, 1975. MR 0385969 | Zbl 0321.60009
[7] W. Nagel: Das Geometrische Kovariogramm und verwandte Größen zweiter Ordnung. Habilitationsschrift, Friedrich-Schiller-Universität Jena (1992).
[8] R. Pyke: Problems corner. IMS Bulletin 18 (1989), 387.
[9] J. Rataj: Characterization of compact sets by their dilation volume. Math. Nachr. 173 (1995), 287–295. DOI 10.1002/mana.19951730116 | MR 1336964 | Zbl 0826.60009
[10] J. Rataj: Estimation of oriented direction distribution of a planar body. Adv. Appl. Probab. 28 (1996), 394–404. DOI 10.2307/1428064 | MR 1387883 | Zbl 0861.60023
[11] J. Rataj: Determination of spherical area measures by means of dilation volumes. Math. Nachr. 235 (2002), 143–162. DOI 10.1002/1522-2616(200202)235:1<143::AID-MANA143>3.0.CO;2-7 | MR 1889282 | Zbl 1005.52004
[12] J. Rataj and M. Zähle: Mixed curvature measures for sets of positive reach and a translative integral formula. Geom. Dedicata 57 (1995), 259–283. DOI 10.1007/BF01263484 | MR 1351855
[13] J. Rataj and M. Zähle: Curvatures and currents for unions of sets with positive reach, II. Ann. Glob. Anal. Geom. 20 (2001), 1–21. DOI 10.1023/A:1010624214933 | MR 1846894
[14] J. Rataj and M. Zähle: A remark on mixed curvature measures for sets with positive reach. Beiträge Alg. Geom. 43 (2002), 171–179. MR 1913777
[15] R. Schneider: On the mean normal measures of a particle process. Adv. Appl. Probab. 33 (2001), 25–38. DOI 10.1239/aap/999187895 | MR 1825314 | Zbl 0978.60013
[16] W. Weil: The estimation of mean shape and mean particle number in overlapping particle systems in the plane. Adv. Appl. Probab. 27 (1995), 102–119. DOI 10.2307/1428099 | MR 1315581 | Zbl 0819.60015
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