Objective: To determine the expression of endogenous adhesion/growth-regulatory lectins and their binding sites using labeled tissue lectins as well as the binding profile of hyaluronic acid as an approach to define new prognostic markers. Methods: Sections of paraffin-embedded histological material of 481 lungs from lung tumor patients following radical lung excision processed by a routine immunohistochemical method (avidin-biotin labeling, DAB chromogen). Specific antibodies against galectins-1 and -3 and the heparin-binding lectin were tested. Staining by labeled galectins and hyaluronic acid was similarly visualized by a routine protocol. After semiquantitative assessment of staining, the results were compared with the pT and pN stages and the histological type. Survival was calculated by univariate and multivariate methods. Results: Binding of galectin-1 and its expression tended to increase, whereas the parameters for galectin-3 decreased in advanced pT and pN stages at a statistically significant level. The number of positive cases was considerably smaller among the cases with small cell lung cancer than in the group with non-small-cell lung cancer, among which adenocarcinomas figured prominently with the exception of galectin-1 expression. Kaplan-Meier computations revealed that the survival rate of patients with galectin-3-binding or galectin-1-expressing tumors was significantly poorer than that of the negative cases. In the multivariate calculations of survival lymph node metastases (p < 0.0001), histological type (p = 0.003), galectin-3-binding capacity (p = 0.01), galectin-3 expression (p = 0.03) and pT status (p = 0.003) proved to be independent prognostic factors, not correlated with the pN stage. Conclusion: The expression and the capacity to bind the adhesion/growth regulatory galectin-3 is defined as an unfavorable prognostic factor not correlated with the pTN stage.

a b s t r a c tIn this paper we develop the theories of normed linear spaces and of linear spaces with indefinite metric, for finite dimensions both of which are also called Minkowski spaces in the literature.In the first part of this paper we collect the common properties of the semi-and indefinite inner products and define the semi-indefinite inner product as well as the corresponding semi-indefinite inner product space. We give a generalized concept of the Minkowski space embedded in a semi-indefinite inner product space using the concept of a new product, which contains the classical cases as special ones.In the second part we investigate the real, finite-dimensional generalized Minkowski space and its sphere of radius i. We prove that it can be regarded as a so-called Minkowski-Finsler space, and if it is homogeneous with respect to linear isometries, then the Minkowski-Finsler distance of its points can be determined by the Minkowski product.

Abstract. In this note we examine the volume of the convex hull of two congruent copies of a convex body in Euclidean n-space, under some subsets of the isometry group of the space. We prove inequalities for this volume if the two bodies are translates, or reflected copies of each other about a common point or a hyperplane containing it. In particular, we give a proof of a related conjecture of Rogers and Shephard.

In this paper we investigate the problem of finding the maximum volume polytopes, inscribed in the unit sphere of the d-dimensional Euclidean space, with a given number of vertices. We solve this problem for polytopes with d + 2 vertices in every dimension, and for polytopes with d + 3 vertices in odd dimensions. For polytopes with d + 3 vertices in even dimensions we give a partial solution.2010 Mathematics Subject Classification. 52B60, 52A40, 52A38.

The concept of angle, angle functions, and the question how to measure angles present old and well-established mathematical topics referring to Euclidean space, and there exist also various extensions to non-Euclidean spaces of different types. In particular, it is very interesting to investigate or to combine (geometric) properties of possible concepts of angle functions and angle measures in finite-dimensional real Banach spaces (= Minkowski spaces). However, going into this direction one will observe that there is no monograph or survey reflecting the complete picture of the existing literature on such concepts in a satisfying manner. We try to close this gap. In this expository paper (containing also new results, and new proofs of known results) the reader will get a comprehensive overview of this field, including also further related aspects. For example, angular bisectors, their applications, and angle types which preserve certain kinds of orthogonality are discussed. The latter aspect yields, of course, an interesting link to the large variety of orthogonality types in such spaces.