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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

This paper deals with a few, not widely known, aspects of Kottman's constant of a Banach space and its symmetric and finite variations. We will consider their behaviour under ultrapowers, relations with other parameters such as Whitley's or James' constant, and connection with the extension of c0-valued Lipschitz maps.1. Kottman's constants. This paper deals with a few, not widely known, aspects of Kottman's constant of a Banach space X, with unit ball B X and unit sphere S X , defined as follows:It was introduced and studied by Kottman in [20,21]. It is clear that K(X) = 0 if and only if X is finite-dimensional. A well-known, although highly non-trivial, result of Elton and Odell [12] (see also [10, p. 241]) establishes that K(X) > 1 for every infinitedimensional Banach space. Kottman's constant has been considered in several papers and its exact calculus in different classical Banach spaces has been performed (see e.g.

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