AN INEQUALITY INVOLVING THE ℓ1, ℓ2, AND ℓ∞ NORMS Academic Article uri icon

abstract

  • Page 1. Analysis and Applications, Vol. 11, No. 6 (2013) 1350026 (5 pages) c© World Scientific Publishing Company DOI: 10.1142/S0219530513500267 AN INEQUALITY INVOLVING THE l1, l2, AND lo NORMS ARYEH KONTOROVICH Department of Computer Science Ben-Gurion University Beer Sheva, Israel karyeh@cs.bgu.ac.il Received 25 September 2012 Revised 12 December 2012 Published 11 October 2013 For x ∈ [0, 1]n with "x"1 = 1 and y ∈ [1, ∞)n, we prove that "xy"∞ "xy"2 ≤ "x"∞"y"1 "y"∞ "y"2 , where xy ∈ Rn is the vector with components xiyi. This bound does not seem to easily follow from known inequalities, and the proof technique may be of independent interest. Keywords: Vector; norm; inequalities. Mathematics Subject Classification 2010: 26D20, 47A30 1. Statement of Main Result Our objective a is to prove Theorem 1.1. For n ∈ N, x ∈ [0, 1] n with ‖x‖1 = 1 and y ∈ [1, ∞) n, we have …

publication date

  • January 1, 2013