Higher Dimensional Conundra

In recent years, especially in the subject of harmonic analysis, there has been interest in geometric phenomena of R as N → +∞. In the present paper we examine several specific geometric phenomena in Euclidean space and calculate the asymptotics as the dimension gets large. 0 Introduction Typically when we do geometry we concentrate on a specific venue in a particular space. Often the context is Euclidean space, and often the work is done in R or R. But in modern work there are many aspects of analysis that are linked to concrete aspects of geometry. And there is often interest in rendering the ideas in Hilbert space or some other infinite dimensional setting. Thus one wants to see how the finite-dimensional result in R changes as N → +∞. In the present paper we study some particular aspects of the geometry of R N and their asymptotic behavior as N → ∞. We choose these particular examples because the results are surprising or especially interesting. One may hope that they will lead to further studies. 1 Volume in R Let us begin by calculating the volume of the unit ball in R and the surface area of its bounding unit sphere. We let ΩN denote the former and ωN−1 denote the latter. In addition, we let Γ(x) be the celebrated Gamma function of L. Euler. It is a helpful intuition (which is literally true when x is an integer) that Γ(x) ≈ (x− 1)!. We shall also use Stirling’s formula which says that k! ≈ k · e · √ 2πk We are happy to thank the American Institute of Mathematics for its hospitality and support during this work.