B-convex space

In mathematics, a B-convex space is a type of Banach space. The concept of B-convexity was related to the strong law of large numbers in Banach spaces by Anatole Beck in 1962; accordingly, it is sometimes referred to as Beck convexity. Beck showed that a Banach space is B-convex if and only if every sequence of independent, symmetric, uniformly bounded and Radon random variables in that space satisfies the strong law of large numbers.

Let X be a Banach space with norm || ||. X is said to be B-convex if for some ε > 0 and some natural number n, it holds true that whenever x₁, ..., x_n are elements of the closed unit ball of X, there is a choice of signs α₁, ..., α_n ∈ {−1, +1} such that

$\left\|\sum _{i=1}^{n}\alpha _{n}x_{n}\right\|\leq (1-\varepsilon )n.$

References

• Beck, Anatole (1962). "A convexity condition in Banach spaces and the strong law of large numbers". Proc. Amer. Math. Soc. 13: 329–334. ISSN 0002-9939. MR0133857
• Ledoux, Michel (1991). Probability in Banach spaces. Berlin: Springer-Verlag. pp. xii+480. ISBN 3-540-52013-9. {{cite book}}: Unknown parameter |coauthor= ignored (|author= suggested) (help) MR1102015 (See chapter 9)