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A concentration of measure phenomenon on uniformly convex bodies. (English) Zbl 0828.52004

Lindenstrauss, J. (ed.) et al., Geometric aspects of functional analysis. Israel seminar (GAFA) 1992-94. Basel: Birkhäuser. Oper. Theory, Adv. Appl. 77, 276-287 (1995).
Let \(B\) be a convex symmetric body in \(\mathbb{R}^d\) with gauge \(|\cdot |_B\), satisfying \[ |x |^p_B + |y |^p_B - 2 \left |{x + y \over 2} \right |^p_B \geq 2 \left |{x - y \over 2K_p} \right |^p_B \quad (x,y \in R^d) \] for some \(p \geq 2\) and \(K_p > 0\), and let \(f : (\partial B, |\cdot |_B) \to R\) be a Lipschitz function with constant \(L\). The fundamental result in this paper states that if \(M\) is a median of \(f\) with respect to \(\mu_{\partial B}\), the normalized relative surface measure of \(B\) with respect to itself given by \[ \mu_{\partial B} (A) = \lim_{\varepsilon \downarrow 0} {\text{Vol}_d \biggl( R^+ A \cap \bigl( (1 + \varepsilon) B \backslash B \bigr) \biggr) \over \text{Vol}_d \bigl( (1 + \varepsilon) B \backslash B \bigr)} \quad (A \text{ measurable in } B), \] then \[ \mu_{\partial B} (f - M > t) \leq 4 \exp \left( - 2d \left( {ct \over K_p L} \right)^p \right) \] for some positive constant \(c\).
For the entire collection see [Zbl 0813.00007].

MSC:

52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
28A75 Length, area, volume, other geometric measure theory
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