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a b s t r a c tWe consider the problem of finding the point in the transportation polytope which is closest to the origin. Recursive formulas to solve it are provided, explaining how they arise from geometric considerations, via projections, and we derive solution algorithms with linear computational complexity in the number of variables.

A subset of projective space is called convex if its intersection with every line is connected. The complement of a projective convex set is again convex. We prove that for any projective convex set there exists a pair of complementary projective subspaces, one contained in the convex set and the other in its complement. This yields their classification up to homotopy.Interest in the interplay between convexity and projective geometry dates back to the early part of the twentieth century (see [5], [7] and [8], or [3] for a survey and applications). More recently, projective methods have played an essential role in the celebrated Karmarkar algorithm for linear programming [6].Let us define a projective convex set as a subset A of real projective space pn, with the property that for every line I (projective, of course), A n I is connected. Clearly, by substituting 'Euclidean' for 'projective' we get back the classic definition of convexity (which, it should be remarked, is an affine notion). But the relation is more than formal.A convex set in Euclidean space remains convex when embedded, through any affine chart, into projective space. Conversely, a convex subset of P n which lies entirely in an affine chart (avoids a hyperplane) is convex there. Thus, projective convex sets generalize Euclidean ones.In the fifties, Dekker [2] and de Groot and de Vries [4] proved that a projective convex set containing no line avoids a hyperplane, thus characterizing Euclidean convex sets among projective ones (beware that we have simplified their terminology). Our main theorem puts this result into a general perspective. REMARK. lf a set A c P~ is convex, so is its complement pn _ A.Indeed, the projective line pl is topologically the circle, and there, if a set is connected so is its complement.Thus, the complement of a point (or a disk) in p2, which is topologically a Moebius Band and a regular neighborhood of a projective line, is convex. This, we shall call a convex set of type 1.DEFINITION. Let A c P~ be convex. The type of A, denoted z(A), is the maximum dimension of a projective subspace of P~ which is contained in A.

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This paper presents a new scheme of applying CORPS (coherently radiating periodic structures) for reducing the number of phase shifters in linear antenna arrays. This scheme can be seen as a combination of the properties of two techniques: CORPS and butler. The proposed system applies an interleaving of several blocks of 2 × 3 CORPS networks. This interleaving of two stages of 2 × 3 CORPS networks is made in a convenient way to provide the required progressive phase for beam-scanning and the level of amplitude excitations necessary for achieving the radiation characteristics of low SLL. Interesting results are provided based on experimental measurements and full-wave simulations to analyze and evaluate the performance of the feeding network based on CORPS and the reduction capability of the number of phase shifters in the antenna system. The proposed design methodology achieves a reduction capability of 66% in the number of phase shifters used in linear antenna arrays. This reduction in the complexity of the antenna system is reached maintaining a peak SLL of −22 dB with scanning ranges of until ±25°. A good design option is provided to simplify the complexity of the feeding network in antenna array applications.