| Title:
|
On set covariance and three-point test sets (English) |
| Author:
|
Rataj, J. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
54 |
| Issue:
|
1 |
| Year:
|
2004 |
| Pages:
|
205-214 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
convex body |
| Keyword:
|
set with positive reach |
| Keyword:
|
normal measure |
| Keyword:
|
set covariance |
| MSC:
|
52A22 |
| MSC:
|
60D05 |
| idZBL:
|
Zbl 1049.52004 |
| idMR:
|
MR2040232 |
| . |
| Date available:
|
2009-09-24T11:11:23Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127877 |
| . |
| Reference:
|
[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. MR 1938112, 10.4310/jdg/1090351101 |
| Reference:
|
[2] H. Federer: Curvature measures.Trans. Amer. Math. Soc. 93 (1959), 418–491. Zbl 0089.38402, MR 0110078, 10.1090/S0002-9947-1959-0110078-1 |
| Reference:
|
[3] H. Federer: Geometric Measure Theory.Springer-Verlag, Berlin, 1969. Zbl 0176.00801, MR 0257325 |
| Reference:
|
[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. |
| Reference:
|
[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 |
| Reference:
|
[6] G. Matheron: Random Sets and Integral Geometry.J. Wiley, New York, 1975. Zbl 0321.60009, MR 0385969 |
| Reference:
|
[7] W. Nagel: Das Geometrische Kovariogramm und verwandte Größen zweiter Ordnung.Habilitationsschrift, Friedrich-Schiller-Universität Jena (1992). |
| Reference:
|
[8] R. Pyke: Problems corner.IMS Bulletin 18 (1989), 387. |
| Reference:
|
[9] J. Rataj: Characterization of compact sets by their dilation volume.Math. Nachr. 173 (1995), 287–295. Zbl 0826.60009, MR 1336964, 10.1002/mana.19951730116 |
| Reference:
|
[10] J. Rataj: Estimation of oriented direction distribution of a planar body.Adv. Appl. Probab. 28 (1996), 394–404. Zbl 0861.60023, MR 1387883, 10.2307/1428064 |
| Reference:
|
[11] J. Rataj: Determination of spherical area measures by means of dilation volumes.Math. Nachr. 235 (2002), 143–162. Zbl 1005.52004, MR 1889282, 10.1002/1522-2616(200202)235:1<143::AID-MANA143>3.0.CO;2-7 |
| Reference:
|
[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. MR 1351855, 10.1007/BF01263484 |
| Reference:
|
[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. MR 1846894, 10.1023/A:1010624214933 |
| Reference:
|
[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 |
| Reference:
|
[15] R. Schneider: On the mean normal measures of a particle process.Adv. Appl. Probab. 33 (2001), 25–38. Zbl 0978.60013, MR 1825314, 10.1239/aap/999187895 |
| Reference:
|
[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. Zbl 0819.60015, MR 1315581, 10.2307/1428099 |
| . |
