Some Elementary Aspects of Means

We raise several elementary questions pertaining to various aspects of means. These questions refer to both known and newly introduced families of means, and include questions of characterizations of certain families, relations among certain families, comparability among the members of certain families, and concordance of certain sequences of means. They also include questions about internality tests for certain mean-looking functions and about certain triangle centers viewed as means of the vertices. The questions are accessible to people with no background in means, and it is also expected that these people can seriously investigate, and contribute to the solutions of, these problems. The solutions are expected to require no more than simple tools from analysis, algebra, functional equations, and geometry. 1. Definitions and Terminology In all that follows, denotes the set of real numbers and denotes an interval in . By a data set (or a list) in a set , we mean a finite subset of in which repetition is allowed. Although the order in which the elements of a data set are written is not significant, we sometimes find it convenient to represent a data set in of size by a point in , the cartesian product of copies of . We will call a data set in ？？ordered if . Clearly, every data set in may be assumed ordered. A mean of variables (or a -dimensional mean) on is defined to be any function that has the internality property for all in . It follows that a mean must have the property for all in . Most means that we encounter in the literature, and all means considered below, are also symmetric in the sense that for all permutations on , and 1-homogeneous in the sense that for all permissible . If and are two -dimensional means on , then we say that if for all . We say that if for all for which are not all equal. This exception is natural since and must be equal, with each being equal to . We say that and are comparable if or . A distance (or a distance function) on a set is defined to be any function that is symmetric and positive definite, that is, Thus a metric is a distance that satisfies the triangle inequality a condition that we find too restrictive for our purposes. 2. Examples of Means The arithmetic, geometric, and harmonic means of two positive numbers were known to the ancient Greeks; see [1, pp. 84–90]. They are usually denoted by , , and , respectively, and are defined, for , by The celebrated inequalities were also known to the Greeks and can be depicted in the well-known figure that is usually attributed to Pappus and that appears in [2, p. 364].