An Approximate Version of the Jordan von Neumann Theorem for Finite Dimensional Real Normed Spaces

It is known that any normed vector space which satisfies the parallelogram law is actually an inner product space. For finite dimensional normed vector spaces over R, we formulate an approximate version of this theorem: if a space approximately satisfies the parallelogram law, then it has a near isometry with Euclidean space. In other words, a small von Neumann Jordan constant E + 1 for X yields a small Banach-Mazur distance with R^n, d(X, R^n) < 1 + B_n E + O(E^2). Finally, we examine how this estimate worsens as the dimension, n, of X increases, with the conclusion that B_n grows quadratically with n.Comment: Version 2 adds contact information for the author and actually states the correct Jordan-von Neumann theorem (oops!

Geometrical constants of Day-James spaces (The generalization of function spaces and its enviroment)

We describe some recent results on the von Neumann-Jordan (NJ-) constant CNJ(X) and the related geometrical constants of concrete Banach spaces X. In particular, we calculate the constants for X being a class of Day-James spaces lp-lq by using the Banach-Mazur distance d(X, H) between X and H, where H is a two-dimensional inner product space

Wheeling around von Neumann–Jordan constant in Banach spaces

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Optimal Weak Parallelogram Constants for L-p Spaces

Inspired by Clarkson\u27s inequalities for L-p and continuing work from [5], this paper computes the optimal constant C in the weak parallelogram laws parallel to f + g parallel to(r )+ C parallel to f - g parallel to(r )= 2(r-1 )(parallel to f parallel to(r) + parallel to g parallel to(r)) for the L-p spaces, 1 \u3c p \u3c infinity

Optimal Weak Parallelogram Constants for LpL^p Spaces

Inspired by Clarkson's inequalities for $L^p$ and continuing work from \cite{CR}, this paper computes the optimal constant $C$ in the weak parallelogram laws $$\|f + g \|^r + C\|f - g\|^r \leq 2^{r-1}\big( \|f\|^r + \|g\|^r \big),$$ $$\|f + g \|^r + C\|f- g \|^r \geq 2^{r-1}\big( \|f\|^r + \|g \|^r \big)$$ for the $L^p$ spaces, $1 < p < \infty$.Comment: 10 page

A characterization of inner product spaces related to the p-angular distance

In this paper we present a new characterization of inner product spaces related to the p-angular distance. We also generalize some results due to Dunkl, Williams, Kirk, Smiley and Al-Rashed by using the notion of p-angular distance.Comment: 9 Pages, to appear in J. Math. Anal. Appl. (JMAA