Average section functions for star-shaped sets. (English) Zbl 1094.52003
Recall that a star-body in Euclidean \(d\)-space \(E^d\) is a compact set which is star-shaped with respect to the origin, contains the origin as an interior point, and has continuous radial function. Let \(K \subset E^d\) be a star body and let \(2 \leq j \leq d-1\).
The paper introduces the average \(s_j(K,u)\) of the intersection volume over all \(j\)-dimensional half-spaces \(H\) containing a given direction \(u\) which is orthogonal to the boundary of \(H\). The authors put the question whether the resulting function \(s_j(K, \cdot)\) on the unit sphere \(S^{d-1}\) determines \(K\) uniquely. They show the uniqueness in the case \(j=2\) for \(d \in \{3, 4 \}\), and in the cases \(j \leq (d+2)/2\) and \(j > (2d+1)/3\) for \(d \geq 5\). They also show infinitely many pairs of the integers \(j\) and \(d\) for which uniqueness fails.
The paper introduces the average \(s_j(K,u)\) of the intersection volume over all \(j\)-dimensional half-spaces \(H\) containing a given direction \(u\) which is orthogonal to the boundary of \(H\). The authors put the question whether the resulting function \(s_j(K, \cdot)\) on the unit sphere \(S^{d-1}\) determines \(K\) uniquely. They show the uniqueness in the case \(j=2\) for \(d \in \{3, 4 \}\), and in the cases \(j \leq (d+2)/2\) and \(j > (2d+1)/3\) for \(d \geq 5\). They also show infinitely many pairs of the integers \(j\) and \(d\) for which uniqueness fails.
Reviewer: Marek Lassak (Bydgoszcz)
MSC:
| 52A30 | Variants of convex sets (star-shaped, (\(m, n\))-convex, etc.) |
| 52A20 | Convex sets in \(n\) dimensions (including convex hypersurfaces) |
| 52A38 | Length, area, volume and convex sets (aspects of convex geometry) |
References:
| [1] | Andrews, G. E.; Askey, R.; Roy, R., Special Functions (1999), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0920.33001 |
| [2] | Baddeley, A. J., Vertical sections, (Ambartzumian, R. V.; Weil, W., Stochastic Geometry, Geometric Statistics, Stereology (1984), Teubner: Teubner Leipzig), 43-52 · Zbl 0568.60012 |
| [3] | Baddeley, A. J.; Gundersen, H. J.G.; Cruz-Orive, L. M., Estimation of surface area from vertical sections, J. Microsc., 142, 259-276 (1986) |
| [4] | Cruz-Orive, L. M.; Howard, C. V., Estimation of individual feature surface area with the vertical spatial grid, J. Microsc., 178, 146-151 (1995) |
| [5] | Fallert, H.; Goodey, P.; Weil, W., Spherical projections and centrally symmetric sets, Adv. Math., 129, 301-322 (1997) · Zbl 0887.52001 |
| [6] | Gardner, R. J., Geometric Tomography (1995), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0864.52001 |
| [7] | Gardner, R. J.; Soranzo, A.; Volčič, A., On the determination of star and convex bodies by section functions, Discrete Comput. Geom., 21, 69-85 (1999) · Zbl 0923.52003 |
| [8] | Ghokale, A. M., Unbiased estimation of curve length in 3-D using vertical slices, J. Microsc., 159, 133-141 (1990) |
| [9] | Goodey, P., Minkowski sums of projections of convex bodies, Mathematika, 45, 253-268 (1998) · Zbl 0959.52003 |
| [10] | Goodey, P., Applications of representation theory to convex bodies, Suppl. Rend. Circ. Mat. Palermo (2), 50, 179-187 (1997) · Zbl 0893.52006 |
| [11] | Goodey, P.; Weil, W., The determination of convex bodies from the mean of random sections, Math. Proc. Cambridge Philos. Soc., 112, 419-430 (1992) · Zbl 0772.52005 |
| [12] | Groemer, H., On a spherical integral transform and sections of star bodies, Monatsh. Math., 126, 117-124 (1998) · Zbl 0918.52002 |
| [13] | Groemer, H., Geometric Applications of Fourier Series and Spherical Harmonics (1996), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0877.52002 |
| [14] | Kiderlen, M., Determination of a convex body from Minkowski sums of its projections, J. London Math. Soc., 70, 2, 529-544 (2004) · Zbl 1069.52003 |
| [15] | Siegel, C. L., Über einige Anwendungen diophantischer Approximationen, (Gesammelte Abhandlungen I (1966), Springer: Springer Berlin), 209-266 (1929), Abh. Preuss. Akad. Wiss., pp. 1-41; reprinted in · JFM 56.0180.05 |
| [16] | Stepanov, S. A., Arithmetic of Algebraic Curves (1994), Consultants Bureau: Consultants Bureau New York · Zbl 0862.11036 |
| [17] | Vedel Jensen, E. B., Local Stereology (1998), World Scientific: World Scientific Singapore · Zbl 0909.62087 |
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