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Onextensions of the Gale-Berlekamp switching problem and constants ofl q —spaces

Authors:	Y Gordon H S Witsenhausen

Institution:	(1) Hebrew University of Jerusalem, Jerusalem;(2) Louisiana State University, Louisiana, USA;(3) Bell Telephone Laboratories, Inc., Murray Hill, N. J.

Abstract:	For positive integersn, m and realp≥1, let $B_p (n, m) = \mathop {\min }\limits_{\varepsilon i j = \pm 1} \mathop {\max }\limits_{\theta _{ i} = \pm 1} \left( {\sum\limits_{j = 1}^m {\| } \sum\limits_{i = 1}^n {\theta _i \varepsilon _{ij} } \|^p } \right)^{1/p}$ Upper and lower bounds for this quantity are derived, extending results of Brown and Spencer forB ₁(n,n), corresponding to the Gale-Berlekamp switching problem. For a Minkowski spaceM of dimensionm, define $\delta (M) = \mathop {\min }\limits_{\|\|x_i \|\| = 1} \mathop {\max }\limits_{\theta _i = \pm 1} \|\|\sum\limits_{i = 1}^m {\theta _i x_i }$ a quantity investigated by Dvoretzky and Rogers.

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