Aljoša Volčič

An algorithm for reconstructing convex bodies from their projections

An algorithm is described for the approximative reconstruction of a plane convex body from its projections in a finite number of directions.A priori anda posteriori error estimates are given, and the convergence of certain sequences of an approximative solution of the reconstruction problem to the exact solution is proven. Finally, it is shown that, after small modifications, the algorithm can be applied to reconstruct convex bodies from discrete projectional data.The algorithm consists in an approximation of the convex body, to be reconstructed, by recursively defined cores and envelopes, following the ideas of Kuba [6] for the reconstruction of binary patterns.

λ-Equidistributed sequences of partitions and a theorem of the De Bruijn-Post type

SummaryThe notion of uniform distribution of a sequence is generalized to sequences of partitions in a separable metric space X. Results concern Riemann integrability with respect to a probability λ on X, and Riemann approximations of Lebesgue integrals.

CONVEX BODIES WITH SIMILAR PROJECTIONS

By examining an example constructed by Petty and McKinney, we show that there are pairs of centered and coaxial bodies of revolution in Ed , d > 3 , whose projections onto each two-dimensional subspace are similar, but which are not themselves even affinely equivalent.

CONVERGENCE IN SHAPE OF STEINER SYMMETRIZATIONS

There are sequences of directions such that, given any compact set KR n , the sequence of iterated Steiner symmetrals of K in these direc- tions converges to a ball. However examples show that Steiner symmetrization along a sequence of directions whose differences are square summable does not generally converge. (Note that this may happen even with sequences of direc- tions which are dense in S n 1 .) Here we show that such sequences converge in shape. The limit need not be an ellipsoid or even a convex set. We also deal with uniformly distributed sequences of directions, and with a recent result of Klain on Steiner symmetrization along sequences chosen from a finite set of directions.

On the Determination of Star and Convex Bodies by Section Functions

Abstract. The i th section function of a star body in

On the well-posedness of the Hammer X-ray problem

{\Bbb E}

Hammer's X-ray problem is well-posed

n gives the i -dimensional volumes of its sections by i -dimensional subspaces. It is shown that no star body is determined among all star bodies, up to reflection in the origin, by any of its i th section functions. Moreover, the set of star bodies that are determined among all star bodies, up to reflection in the origin, by their i th section functions for all i , is a nowhere dense set. The determination of convex bodies in this sense is also studied. The results complement and contrast with recent results on the determination of convex bodies by i th projection functions. The paper continues the development of the dual Brunn—Minkowski theory initiated by Lutwak.

When do sections of different dimensions determine a convex body

SummaryIn un precedente articolo [10] il secondo autore ha dimostrato che ogni corpo convesso piano è univocamente determinato dalle lunghezze dette corde tagliate da tre fasci propri, i cui sostegni non sono allineati. Si dimostra in questo articolo che il corrispondente problema ricostruttivo è ben posto, se si assume che il corpo convesso da ricostruire è contenuto in un preassegnato insieme limitato.

On the Bertrand paradox

SummaryIn un precedente articolo [2]G. Mägerl ed il secondo autore hanno dimostrato la buona posizione del problema ricostruttivo di Hammer, imponendo che il corpo convesso da ricostruire fosse contenuto in un preassegnato insieme limitato. Si dimostra qui che tale limitazione può essere tolta.

A Three-Point Solution to Hammer's X-Ray Problem

This paper gives a partial answer to a problem posed by Volcic and shows, in particular, that a three-dimensional convex body K is uniquely determined if p ′ and p ″ are two points interior to K and the lengths of all the chords of K through p ′ and the areas of all sections of K with planes through p ″ are known, provided that a specific condition on the positions of p ′ and p ″ with respect to K is satisfied. The problem will be studied in the more general framework of i -chord functions, and the results will also cover cases where the points p ′ and p ″ are not interior to K , possibly with one of them at infinity.