We deal with the question of uniqueness, namely to decide when an unknown finite set of points in Z 2 is uniquely determined by its X-rays corresponding to a given set S of lattice directions. In Hajdu (2005) [11] proved that for any fixed rectangle A in Z 2 there exists a non trivial set S of four lattice directions, depending only on the size of A, such that any two subsets of A can be distinguished by means of their X-rays taken in the directions in S. The proof was given by explicitly constructing a suitable set S in any possible case. We improve this result by showing that in fact whole families of suitable sets of four directions can be found, for which we provide a complete characterization. This permits us to easily solve some related problems and the computational aspects
Fencing problems regard the bisection of a convex body in a way that some geometric measures are optimized. We introduce the notion of relative diameter and study bisections of centrally symmetric planar convex bodies, bisections by straight line cuts in general planar convex bodies and also bisections by hyperplane cuts for convex bodies in higher dimensions. In the planar case we obtain the best possible lower bound for the ratio between the relative diameter and the area. 2004 Elsevier Inc. All rights reserved.
Abstract. In Discrete Tomography there is a wide literature concerning (weakly) bad configurations. These occur in dealing with several questions concerning the important issues of uniqueness and additivity. Discrete lattice sets which are additive with respect to a given set S of lattice directions are uniquely determined by X-rays in the direction of S. These sets are characterized by the absence of weakly bad configurations for S. On the other side, if a set has a bad configuration with respect to S, then it is not uniquely determined by the X-rays in the directions of S, and consequently it is also non-additive. Between these two opposite situations there are also the non-additive sets of uniqueness, which deserve interest in Discrete Tomography, since their unique reconstruction cannot be derived via the additivity property. In this paper we wish to investigate possible interplays among such notions in a given lattice grid A, under X-rays taken in directions belonging to a set S of four lattice directions.
Integral representations are obtained of positive additive functionals on finite products of the space of continuous functions (or of bounded Borel functions) on a compact Hausdorff space. These are shown to yield characterizations of the dual mixed volume, the fundamental concept in the dual Brunn-Minkowski theory. The characterizations are shown to be best possible in the sense that none of the assumptions can be omitted. The results obtained are in the spirit of a similar characterization of the mixed volume in the classical Brunn-Minkowski theory, obtained recently by Milman and Schneider, but the methods employed are completely different.
In this paper we use the algebraic approach to Discrete Tomography introduced by Hajdu and Tijdeman to study functions f : Z 2 → {−1, 0, +1} which have zero line sums along the lines corresponding to certain sets of four directions.
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