On the Dual Form of ‘Low M * Estimate’ in the Quasi-convex Case

On the Dual Form of ‘Low M * Estimate’ in the Quasi-convex Case Let $$B_{2}^{n}$$ denote the Euclidean ball in $${\mathbb R}^n$$ , and, given closed star-shaped body $$K \subset {\mathbb R}^{n}, M_{K}$$ denote the average of the gauge of K on the Euclidean sphere. Let $$p \in (0,1)$$ and let $$K \subset {\mathbb R}^{n}$$ be a p-convex body. In [17] we proved that for every $$\lambda \in (0,1)$$ there exists an orthogonal projection P of rank $$(1 - \lambda)n$$ such that $$\frac{f(\lambda)}{M_K} PB^{n}_{2} \subset PK,$$ where $$f(\lambda)=c_p\lambda^{1+1/p}$$ for some positive constant c p depending on p only. In this note we prove that $$f(\lambda)$$ can be taken equal to $$C_p\lambda^{1/p-1/2}$$ . In terms of Kolmogorov numbers it means that for every $$k \leq n$$ $$d_k (\hbox{Id}:\ell^{n}_{2} \to ({\mathbb R}^{n},\|\cdot\|_{K})) \leq C_p \frac{n^{1/p-1}}{k^{1/p-1/2}} \ell (\hbox{ID}: \ell^{n}_{2} \to ({\mathbb R}^{n}, \|\cdot\|_{K})),$$ where $$\ell(\hbox{Id})={\bf E}\|\sum\limits^{n}_{i=1}g_i e_i\|_K$$ for the independent standard Gaussian random variables $$\{g_i\}$$ and the canonical basis $$\{e_i\}$$ of $${\mathbb R}^n$$ . All results do not require the symmetry of K. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

On the Dual Form of ‘Low M * Estimate’ in the Quasi-convex Case

Positivity , Volume 8 (4) – May 7, 2003

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Kluwer Academic Publishers
Copyright © 2005 by Kluwer Academic Publishers
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
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