Statistical sampling and fractal distributions

We report here on work that has its origin in a problem of statistical sampling and that has extended over more than a few years. There are connections with several other areas of mathematics, including convex analysis, tomography, and, perhaps more surprisingly, fractal geometry and iterated function systems. This summary stresses these connections and the more visual aspects of the work. To introduce the ideas we direct the reader's attention to Figure 1. There we see a complicated starlike object sitting inside the regular hexagon. As we shall see, each point in the hexagon encodes three (dependent) coordinates Xk summing to 0. These coordinates provide a set of sampling values that are exactly "balanced"; in the second section of this article we shall say something about the statistical advantages of choosing the balanced sampling points according to a particular distribution within the hexagon, such as that shown in Figure 1. The "superstar" appearing in Figure 1 is a self-similar fractal in the sense of Mandelbrot [M]; it has dimension In 6/ ln 3 ~ 1.631; and it is riddled with infinitely many jagged holes, each bounded by the classic Koch snowflake curve. The superstar inhabits a regular hexagon which we denote by M(3) in this article; think of 3 as the sample size of the corresponding statistical procedure. The superstar may be generated by an iterated function system (IFS), in the sense of Barnsley [B], involving the six maps that shrink (with contraction factor ~) the hexagon M(3) toward its six vertices. The superstar may also be viewed as the support of an uncountable family of extreme points in a certain convex body of probability measures. We'll discuss these matters in more detail a little later. The histogram at the base of Figure 1 records the frequency distribution of the horizontal component of