Talk:Modulus and characteristic of convexity
 | This article is rated Start-class on Wikipedia's content assessment scale. It is of interest to the following WikiProjects: | ||||||||||
| |||||||||||
[Untitled]
[edit]On my addition just made:
I have changed the definition of modulus of convexity to the form more commonly used, where the infimum is taken over pairs of points in the unit _sphere_ and not the unit _ball_. This is the formulation originally used by Day, and in most textbook treatments of the subject (including all of the references that previously existed, with the exception of the paper of Goebel). As it is not a priori clear that the two formulations give the same result--- if indeed they do give the same result, and that isn't clear to me--- it seems important that this page reflect the more popular definition in the literature. (Goebel, and a handful of papers citing Goebel, are the only people who use the unit ball in that definition, as far as I can tell.)
I have removed the reference to an apparent disagreement in the literature about whether or not the modulus of convexity must itself be convex. It seems to me that there is no disagreement: there are several published examples of spaces whose moduli of convexity are not convex. I do not know what to make of Goebel's assertion that it must be convex, but here are my thoughts:
First, Goebel's definition, with the unit ball in place of the unit sphere, may well give a different function that _is_ convex. But if this is the case, his "modulus of convexity" is not the function that the majority of the literature (in particular, the original paper of M.M.Day in which the term was defined) refers to by that name, so it should not be the main reference for this entry.
Second, Goebel's proof of the convexity of the modulus of convexity is not at all transparent. For example:
(1) It contains the assertion that "the [implicitly pointwise] infimum of an arbitrary family of convex functions is convex", which is patently false, for example, if f is the exponential function on the real line, and g(x) = f(-x), then f and g are convex but their pointwise infimum is the function e^(-|x|) which is not convex. A statement like this could certainly be true if more structure is assumed of the family, but an _arbitrary_ infimum of convex functions does not have to be convex!
(2) Immediately following the definitions of N(u,v) and the function delta(u,v,epsilon), he says "let (x_1, y_1) be in N(u,v), and (x_2, y_2) be in N(u,v), such that the norm of x_1 - y_1 is at least epsilon_1 and the norm of x_2 - y_2 is at least epsilon_2. Put x_3 = (x_1 + x_2)/2 and y_3 = (y_1 + y_2)/2. Obviously (x_3,y_3) is in N(u,v) and hence the norm of x_3 - y_3 is [...] at least epsilon_1 + epsilon_2." This is patently false, as one can see by fixing any Banach space to start, and any nonzero u and v, and taking epsilon_1 and epsilon_2 to be the norm of u, and x_1 = u, y_1 = 0, x_2 = -u, and y_2 = 0. It them happens that x_3 and y_3 as Goebel defines them are both 0, and this is not at least epsilon_1 + epsilon_2.
I am not saying that Goebel's general idea is not fixable--- it could be that it is--- but as it is not presented clearly, it should not be given as a reference for the modulus of convexity. (If indeed Goebel's modulus of convexity does coincide with the standard one referred to in the literature.) 173.28.182.22 (talk)