Triangles Inscribed in a Semicircle, in Minkowski Planes, and in Normed Spaces

Abstract

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

Estimates for Kottman's separation constant in reflexive Banach spaces

“…(i 1 ) in the definition of K(X), we can replace S X with B X ; (i 2 ) for any infinite-dimensional space we have K(X) > 1 (this is a deep result proved in [5]); the range of K(X), even if we restrict ourselves to the class of reflexive spaces, is (1,2] (see [12, p. 21]);…”

On the Baronti constants of Orlicz function spaces

“…The inequality J(X) A 2 (X) follows from (1) and (2). By the Cauchy inequality and (3) we have for x, y ∈ S(X)…”

Geometric mean and triangles inscribed in a semicircle in Banach spaces

“…It is well known (see, e.g., [9] and [4]) that, for 1 p 2, j ( p ) = A 1 ( p ) = 2 1/p . Hence, from (2) it follows that also t ( p ) = 2 1/p .…”

“…The relationship of these and other similar constants with important geometric properties of the space such as, for example, its normal structure (and hence fixed point property) has been extensively studied (see [4,10] and [15]). These constants are also closely related to the well-known Clarkson modulus of convexity, as we shall recall later.…”

“…In [4,5,9,10,[15][16][17] one can find further information about the constants defined above. In particular, it is proved in [17] that in the definition of J (X), S(X), G(X) and j (X) we can consider only pairs x, y such that x + y = x − y .…”