Nonlinear Studies - www. nonlinearstudies.com MESA - www.journalmesa.com Preprint submitted to Nonlinear Studies / MESA On geometric construction of some power means Ralph Høibakk 1 , Dag Lukkassen 1,2 , Lars-Erik Persson 1 , Annette Meidell 1,2 ⋆ 1 UiT The Arctic University of Norway, Lodve Langesgate 2, N8505 Narvik, Norway 2 NORUT Narvik 2 , Rombaksveien 47, N8517 Narvik, Norway ⋆ Corresponding Author: E-mail: annette.meidell@uit.no Abstract. In the homogenization theory, there are many examples where the effective conductivities of composite structures are power means of the local conductivities. The main aim of this paper is to initiate research concerning geometric construction of some power means of three or more variables. We contribute by giving methods for the geometric construction of the harmonic mean P −1 and the arithmetic mean P 1 of three variables a, b and c 1 Introduction As it is well known, homogenization theory, as well as PDEs, plays pretty important roles in the study of many applied problems, see [7]. The use of power means is of certain interest in homog- enization theory. This is natural since the this theory is mainly handling differential equations with rapidly oscillating coefficients. These equations can be replaced with a homogenized equation where the coefficients can be interpreted as special means. Therefore many research papers are devoted to development of methods, say, tools for such theories. Sometimes just a simple inequality or correctly discovered relation between some parameters can be extremely useful for solving problems, where it was unclear how one can find an appropriate approach. For instance, in the study of a scale of two-component composite structures of equal proportions with infinitely many micro-levels, it was found that their effective conductivities are power means of the local conductivities, see [8] and [9]. As regarding to such a basic concept as power means and its relation with important Jensen’s inequality, we refer e.g. to the new book [10], see also references therein. Power means have fascinated mathematicians during many centuries. In the simplest form they can be described as follows: For n positive numbers, (a 1 , a 2 , ....., a n ), the power mean P n k of order k with equal weights is defined as: 2010 Mathematics Subject Classification: 03C25, 33F05, 65D20 Keywords:Power means, geometric construction, homogenization theory.