Convexity is an old idea that goes back to Archimedes's investigations of area and length. See Figure 1. It was studied sporadically over the century by Cauchy, Lagrange, and others. But the subject area was not actually formalized until the work of Minkowski, Steiner, Funk, Radon, and others in the nineteenth and early twentieth centuries.

Figure 1

Today convexity is a vital area in many parts of mathematics, ranging from real analysis and functional analysis to geometry, optimization theory, and many other parts of analytical thinking. This workshop concentrated on the study of convex bodies.

The geometry of convex bodies is generally based on the analysis of cross-sections and also of projections. In particular, there are new ways to approach these ideas using Fourier analysis. Fundamental geometric properties of convex bodies may be expressed Fourier-analytically, and thus Fourier-analytic tools may be used to attack geometric questions.

One matter of interest is to study hyperplane sections of the unit cube Q_N in R^N. Pierre Laplace (1749--1827) studied the area of the central hyperplane section S perpendicular to the main diagonal. See Figure 2. He produced the formula

Vol(Q_N ∩ S) = (1/π) ∫_-∞^∞ ( [sin(r/√N)]/[r/√N] )^N dr .

Figure 2

Interestingly, this last integral is precisely the formula that appears in the central limit theorem for the sums of uniformly distributed random variables (a subject that would not exist for another 150 years after Laplace). Moreover, one may calculate that the volume of this diagonal section tends to the limit 6/√π as N → ∞. Certainly the asymptotics of results in convex geometry as the dimension N → ∞ is a matter of great interest. This is true in part because such results will shed light on what is true in infinite dimensional spaces, such as Hilbert space.

There is actually a remarkable surprising lurking in the background of the preceding discussion. The figure (which is only 3-dimensional!) suggests, and our intuition reaffirms, that the (N-1)-dimensional slice S is the slice of the cube having greatest area. This turns out to be false. In fact in every dimension the slice perpendicular to (1,1, 0, ..., 0) has volume √2, and that exceeds 6/√π. See Figure 3.

Figure 3

Of course any new result spawns new questions and new directions, and so it was with this last remarkable result. It became a problem of great interest to find the hyperplane section of the cube in R^N having greatest (N-1)-dimensional area. Hensley, in 1979, showed that there is an upper bound on the area of a slice---and that bound is independent of dimension. Keith Ball, in 1986, showed that the sharp upper bound is in fact √2.

Both these results used a remarkable fact discovered by Georg Polya in 1913: If ξ is a unit vector in R, and ξ^⊥ is the central hyperplane perpendicular to ξ, then

Vol (Q_N ∩ ξ^⊥) = (1/π) ∫_-∞^∞ ∏_k=1^N (sin r ξ_k)/[r ξ_k] dr .

The proof of this formula involves the Fourier transform (see below).

The Fourier approach to convex geometry played an important role in the solution of the Busemann-Petty problem, one of the most significant questions in the area. It was posed in 1956 and finally solved at the end of the 1990s. It is as follows: \begin{quote} Suppose that K and L are origin-symmetric convex bodies in R so that

Vol (K ∩ ξ^⊥) ≤ Vol (L ∩ ξ^⊥)

for all unit vectors ξ ∈ R^N. Does it follow that Vol (K) ≤ Vol (L)?

It turns out---rather dramatically---that the solution to this problem is affirmative in dimensions less than or equal to 4 and negative in dimensions greater than 4. A great many mathematicians played a role in the solution of this problem. The solution came forth incrementally over a period of more than twenty years. Zvavitch (one of the organizers) has produced remarkable generalizations of the problem.

We have alluded several times now to the Fourier transform without saying exactly what it is. If f is an integrable function on R^N then we define the Fourier transform of f to be

F(s) = ∫_R^N f(x) e^{i x ⋅ s} dx .

This important operation decomposes the given f into frequencies that contain important information about analytic and geometric properties of f. A great deal of modern analysis consists in studying the Fourier transform and its subtle characteristics. Much of modern signal processing and image analysis---among many other areas---depends on the Fourier transform and cognate ideas.

The important Fourier inversion formula (valid when both f and F are integrable)

f(x) = c_N ⋅ ∫_R^N F(s) e^{-i x ⋅ s} ds

tells us that the original function f may be recovered from its Fourier transform.

The workshop on Fourier Analytic Methods in Convex Geometry explored questions such as we have described here using the Fourier transform and allied tools. The investigations were wide-ranging, and have applications ranging from signal processing to computer graphics.