In this paper we mainly consider triangles inscribed in a semicircle of a normed space; in two-dimensional spaces, their perimeter has connections with the perimeter of the sphere. Moreover, by using the largest values the perimeter of such triangles can have, we define two new, simple parameters in real normed spaces: one of these parameters is strictly connected with the modulus of convexity of the space, while the study of the other one seems to be more complicated. We calculate the value of our two parameters and we bring out a few connections among their values and the geometry of real normed spaces. ᮊ

Abstract. In this paper we study the w * -fixed point property for nonexpansive mappings. First we show that the dual space X * lacks the w * -fixed point property whenever X contains an isometric copy of the space c. Then, the main result of our paper provides several characterizations of weak-star topologies that fail the fixed point property for nonexpansive mappings in ℓ 1 space. This result allows us to obtain a characterization of all separable Lindenstrauss spaces X inducing the failure of w * -fixed point property in X * .

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Abstract. The main aim of the present paper is to investigate various structural properties of hyperplanes of c, the Banach space of the convergent sequences. In particular, we give an explicit formula for the projection constants and we prove that an hyperplane of c is isometric to the whole space if and only if it is 1-complemented. Moreover, we obtain the classification of those hyperplanes for which their duals are isometric to ℓ 1 and we give a complete description of the preduals of ℓ 1 under the assumption that the standard basis of ℓ 1 is weak * -convergent.