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In recent years my research has been in a new field called geometric tomography, an area of mathematics dealing with the retrieval of information about a geometric object from data concerning its projections ("shadows") on planes and/or sections by planes. The subject has connections with convex geometry, stereology, geometric probing in robotics, computerized tomography, and other areas. The second edition of my 1995 book "Geometric Tomography" was published by Cambridge University Press (New York) in June 2006, and is available in both hardback
and
paperback. It is designed to be somewhat accessible even to advanced undergraduate students, and contains 79 computer-generated pictures, 66 open problems, and 872 references.
I acknowledge the support of the National Science Foundation in my research on geometric tomography.
| Related Links
Workshop on Geometric Tomography, Alicante, Spain, October 5 to 10, 2004.
Second Summer School on Stereology and Geometric Tomography, Sandbjerg Estate, Denmark, June 17 to 21, 2002.
First Summer School on Local Stereology and Geometric Tomography, Sandbjerg Estate, Denmark, May 20 to 25, 2000.
Another
Geometric Tomography web page with a nice animation of 3-dimensional sections of a 4-dimensional cube.
Discrete Tomography is a related area with a life of its own.
Workshop on Discrete Tomography and Its Applications, New York, June 13 to 15, 2005.
There is a large overlap between geometric tomography and convex geometry. See Paolo Gronchi's web page for a nice group photo of participants at the 2003 meeting on the analytical aspects of convex geometry at Cortona, Italy.
Workshop on Geometric Inequalities, Florence, Italy, May 16 to 20, 2005.
| Recent Papers
Some auxiliary material for published papers:
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An early version of my paper
The Brunn-Minkowski inequality, Bull. Amer. Math. Soc.
39 (2002), 355-405 that contains several full proofs.
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The proof of estimate (7) in
R. J. Gardner and Peyman Milanfar, Reconstruction of convex bodies from brightness functions, Discrete Comput. Geom.
29 (2003), 279-303.
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An extended version of:
R. J. Gardner, Markus Kiderlen, and Peyman Milanfar, Convergence of algorithms for reconstructing convex bodies and directional measures, Ann. Statist. 34 (2006), 1331-1374.
Papers just published:
Paolo Dulio, R. J. Gardner, and Carla Peri, Discrete point X-rays, SIAM J. Discrete Math. 20 (2006), 171--188.
Amyn Poonawala, Peyman Milanfar, and R. J. Gardner, Shape estimation from support and diameter functions, J. Math. Imaging Vision 24 (2006), 229--244.
R. J. Gardner, Markus Kiderlen, and Peyman Milanfar, Convergence of algorithms for reconstructing convex bodies and directional measures, Ann. Statist. 34 (2006), 1331-1374.
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R. J. Gardner and Markus Kiderlen, A solution to Hammer`s X-ray reconstruction problem, Adv. Math.
214 (2007), 323-343.
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R. J. Gardner, The dual Brunn-Minkowski theory for bounded Borel sets: Dual affine quermassintegrals and inequalities, Adv. Math.
216 (2007), 358-386.
Preprints:
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R. J. Gardner and Artem Zvavitch, Gaussian Brunn-Minkowski inequalities, submitted.
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R. J. Gardner and Markus Kiderlen, A new algorithm for 3D reconstruction from support functions, submitted.
To see a full list of publications, consult my curriculum vitae
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or the excellent MathSciNet search engine, where reviews and citations
can also be found.
Richard J. Gardner's home page