Abstract
The parallel X-ray of a convex set K⊂ℝn in a direction u is the function that associates to each line l, parallel to u, the length of K∩l. The problem of finding a set of directions such that the corresponding X-rays distinguish any two convex bodies has been widely studied in geometric tomography. In this paper we are interested in the restriction of this problem to convex cones, and we are motivated by some applications of this case to the covariogram problem. We prove that the determination of a cone by parallel X-rays is equivalent to the determination of its sections from a different type of tomographic data (namely, point X-rays of a suitable order). We prove some new results for the corresponding problem which imply, for instance, that convex polyhedral cones in ℝ3 are determined by parallel X-rays in certain sets of two or three directions. The obtained results are optimal.
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References
Averkov, G., Bianchi, G.: Confirmation of Matheron’s conjecture on the covariogram of a planar convex body. Preprint
Bianchi, G.: Matheron’s conjecture for the covariogram problem. J. Lond. Math. Soc. (2) 71(1), 203–220 (2005)
Bianchi, G.: The covariogram determines three-dimensional convex polytopes. Preprint
Dulio, P., Longinetti, M., Peri, C., Venturi, A.: Sharp affine stability estimates for Hammer’s problem. Adv. Appl. Math. (in press)
Falconer, K.J.: Hammer’s X-ray problem and the stable manifold theorem. J. Lond. Math. Soc. (2) 28(1), 149–160 (1983)
Falconer, K.J.: X-ray problems for point sources. Proc. Lond. Math. Soc. (3) 46(2), 241–262 (1983)
Gardner, R.J.: Symmetrals and X-rays of planar convex bodies. Arch. Math. (Basel) 41(2), 183–189 (1983)
Gardner, R.J.: Chord functions of convex bodies. J. Lond. Math. Soc. (2) 36(2), 314–326 (1987)
Gardner, R.J.: X-rays of polygons. Discrete Comput. Geom. 7(3), 281–293 (1992)
Gardner, R.J.: Geometric Tomography, 2nd edn. Encyclopedia of Mathematics and its Applications, vol. 58. Cambridge University Press, Cambridge (2006)
Gardner, R.J., Kiderlen, M.: A solution to Hammer’s X-ray reconstruction problem. Adv. Math. 214(1), 323–343 (2007)
Gardner, R.J., McMullen, P.: On Hammer’s X-ray problem. J. Lond. Math. Soc. (2) 21(1), 171–175 (1980)
Lam, D., Solmon, D.C.: Reconstructing convex polygons in the plane from one directed X-ray. Discrete Comput. Geom. 26(1), 105–146 (2001)
Michelacci, G.: On a partial extension of a theorem of Falconer. Ric. Mat. 37(2), 213–220 (1988)
Mani-Levitska, P.: Unpublished note (2001)
Natterer, F.: The Mathematics of Computerized Tomography. Teubner, Stuttgart (1986)
Volčič, A.: A three-point solution to Hammer’s X-ray problem. J. Lond. Math. Soc. (2) 34(2), 349–359 (1986)
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Bianchi, G. Geometric Tomography of Convex Cones. Discrete Comput Geom 41, 61–76 (2009). https://doi.org/10.1007/s00454-008-9061-2
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DOI: https://doi.org/10.1007/s00454-008-9061-2