Every norm is not logarithmically convex

Abstract

This correspondence relates to the remark in a recent paper by D.G. Luenberger [ibid., vol. SSC-4, pp. 182-188, July 1968] that any norm defined on a vector space is a real convex function. Although this is a well-known fact in mathematics, a less well-known fact is that every logarithmically convex function is positive and convex, but not conversely, i.e., there are positive convex functions which are not logarithmically convex. As the above title indicates, norms are such functions. This mathematical remark relates to systems science through several areas of application where logarithmic convexity is a highly useful property. In particular, Klinger and Mangasarian ["Logarithmic convexity and geometric programming," J. Math. Anal. and Appl., vol. 24, pp. 388-408, November 1968] mention optimization of multiplicative criteria, reliability theory, and electrical network synthesis, and examine geometric programming in detail.