Università degli Studi della Calabria

Monday, May 17, 2010 10:08

Author of over 50 publications in measure theory (Radon-Nikodym theorem, equidistributed sequences), topology (sequential spaces and fixed point theorems). Some relevant contributions in convexity (determination of convex bodies from measures of sections or projections) and the covariogram problem, with applications to local stereology. and random Steiner symmetrizations. More recently I am developing, with several collaborators, a theory of uniformly distributed sequences of partitions.

My main results are cited in at least eleven books: the volume of R.J. Gardner, "Geometric Tomography", Cambridge Univ. Press, 1995, second edition 2006, "Handbook of Convex Geometry", edited by P.M. Gruber and J.M. Wills, North Holland, 1993 the two books by A.C. Thompson "Minkowski Geometry" and by H. Groemer, "Geometric Applications of Fourier Analysis and Spherical Harmonics", both published by Cambridge Univ. Press 1996, the book by M. Drmota and R. F. Tichy, Sequences, discrepancies and applications", Lecture Notes in Mathematics 1651, Springer 1997, the “Handbook of Measure Theory”, Elsevier 2002 (and not only in the chapter written by me with D. Candeloro), and then expository books like the one by V. Klee and S. Wagon, "Unsolved Problems in Plane Geometry", Dolciani Math. Expositions, MAM, 1991 and the other one by H.T. Croft, K.J. Falconer and R.K. Guy "Unsolved Problems in Geometry ", Problem Books in Mathematics, Springer, 1991, and also some books devoted to applications of mathematics, like the by now classical book by F. Natterer, "The Mathematics of Computerized Tomography", Teubner, 1986, and the volumes edited by J. R. Goodman and J. O'Rourke "Handbook of Computational Geometry", CRC Press, 1997 and the volume "Discrete Tomography, Fundations, Algorithms and Applications", edited by A. Kuba and G. T. Herman, Birkhauser 1999.

The main results I have obtained are the following:

In measure theory I extended the Radon-Nikodym-Segal theorem. This version is in the Handbook of Measure Theory, Elsevier 2002. It gets rid of any assumption on the relation between the two measures in play. Under these conditions, Q-singularity between “pieces” of the two measures appears, a typical situation which can be observed when dealing with Hausdorff measures of different dimension. The aim of this study was the attempt to check wether the Hausdorff measures of positive dimension and having dimension smaller than the dimension of the euclidean space are Maharam (called also localizable). The problem is still open. In connection with this problem I was also interested in the theory of lifting, a concept introduced on the real line by von Neumann and in the abstract setting by D. Maharam. The theme was “modern” at that time, after the publication in ’71 of the book by A. and C. Ionescu-Tulcea "Topics in the theory of liftings".
I extended the concept of lifting to the Daniell integral. I also solved three (out of fifteen) problems posed in the book "Differentiation von Massen by D. Koelzow, Lecture Notes in Mathematics 65, Springer (1968).

In convex geometry I contributed to solve Hammer’s X-ray problem (posed in 1961) on the riconstruction of convex bodies from a finite number of X-ray pictures. The most relevant result has been published in "A three-point solution to Hammer's X-ray problem. J. London Math. Soc. (2) 34 (1986), no. 2, 349-359". The book by R.J. Gardner quoted above cites in the references about twenty papers of mine and of mine collaborators and devotes about fifty pages to the description of my results.

I was also interested in the problem of reconstructing matrices with entries in {0,1} from row and columns sums, establishing a connection between the discrete and the continuous case of reconstructing a convex (and more generally a bounded measurable) set from two X-ray pictures. References to that papers can be found in the volumes edited by J. E. Goodman and J. O'Rourke and, respectively, by A. Kuba and G. T. Herman (Chapters 1, 5 and 6). To the same circle of ideas belongs a positive answer (in collaboration with T. Zamfirescu) to a conjecture posed verbally by P. Gruber in a meeting held in 1984 at the Institute of Oberwolfach. Later the same question was written explicitly in the introduction to the collected papers of J. Radon, edited by P. Gruber. We proved that “almost all” the n-dimensional convex bodies (in the sense of Baire categories) are determined by just two X-ray pictures.

The so-called “equichordal problem” by Fujiwara, Blaschke, Rothe and Weitzenböck caught my attention in the late eighties. My result (with G. Michelacci, A better bound for the excentricities not admitting the equichordal body. Arch. Math. (55) (1990), no. 6, 599-609), complemented by the paper by R. Schafke and H. Volkmer (Asymptotic analysis of the equichordal problem, J . Reine Angew. Math. 425 (1992), 9–60) was considered for a while by the experts the most promising way to solve (in the negative) the classical problem posed in 1916. This attempt has been obviously abbandoned after the announcement of the solution, obtained with completely different method, by M. Rychlik (A Complete Solution to the Equichordal Problem of Fujiwara, Blaschke, Rothe, and Weitzenböck. Invent. Math. 129, 141-212, 1997).

I gave a solution to the so-called "first digit problem", motivated by a statement contained in the book by D. Knuth, The Art of Computer Programming, asserting (vol.2, pp. 219-229) that the deduction of the Benford distribution from the scale-invariance, contained in a paper by R. S. Pinkham, was not correct. My paper appeared unfortunately few months after Theodor P. Hill, when i was still writing the final version of the paper, published his proof (essentially equivalent to mine), so the priority was attibuted justly to him.

The paper with G. Bianchi and F. Segala "The solution of the covariogram problem for plane C2_+ convex bodies", J. Differential Geom, 60 (2002), no.2, 177-198 solved the covariogram problem (posed by Matheron in 1986) for planar convex bodies under the assumption of continuous and positive curvature. That result was a start for G. Bianchi who solved the problem completely.

Recently I am working on a three-dimensional problem of geometric tomography. The initial problem has been solved and now we want to see what can the new methods say in higher dimension, for which the literature is still rather poor.

A problem from convex geometry (due to Mani-Levitska, posed in 1986) motivated the study on the convergence of Steiner symmetrizations, showing that almost surely a random sequence of Steiner symmetrizations (chosen according to the uniform law) of any bounded measurable set in n-dimensional space converges with respect to the symmetric-difference distance to the ball having the same volume. The result has been extended to L_p convergence of functions and the latter extension allowed, in collaboration with A. Burchard, to solve Mani's problem.

In the last years I began to build the bases for a theory of uniformly distributed (u.d.) sequences of partitions of [0,1]. The first result is due to S. Kakutani (in response to a question posed by the physicist H. Araki). In a bunch of papers –some published, some submitted- I introduced a generalization of Kakutani’s construction on [0,1], proving the uniform distribution of a much wider class of sequences. I extended (in collaboration with I. Carbone) the construction to higher dimendion, with a method which is intrinsically pluri-dimensional. I showed moreover that, with probability 1, the sequences of points which are associated in a natural manner to a u.d. sequence of partitions are u.d.. I am finishing a paper which extends some results to fractals which satisfy the Open Set Condition.

Cubo 31B III piano

Dipartimento di Matematica e Informatica Via P. Bucci - Cubo 30B - ponte stradale Arcavacata di Rende (CS)
University of Calabria

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