Convex geometry is an old subject, finding its roots in considerations by Archimedes of arc length. The idea was considered over time by Cauchy, Lagrange, and others. But it is only in the twentieth century that the subject area had been formalized and isolated. The treatise Theorie der konvexen Körper by Bonneson and Fenchel is one of the first treatises on complex geometry.

And there are many classical questions that are fundamental parts of the subject, and that continue to influence our thinking today. One of these is the isoperimetric problem. Stated in a basic form, this question asks which planar region of perimeter 2π has greatest area. See Figure 1.

Figure 1

Of course the same question may be asked in Euclidean space of any dimension, with "perimeter" replaced by "boundary measure" and "area" replaced by "volume".

The classical answer in the plane to the isoperimetric problem is that the disc of radius 1 has greatest area. In fact one may sharpen the result with the isoperimetric inequality:

A ≤ p²/4π

(where A is area and p is perimeter). Analogous results hold in higher dimensions.

Some of the questions considered by the current workshop are inspired by the isoperimetric inequality. An example is to consider regions in the sphere (the set ∑_j x_j² = 1 in n-dimensional Euclidean space) having area 1/2. Which of these has boundary of least area (or length)? See Figure 2, which illustrates the idea for the sphere in Euclidean 3-space. Then the region we are considering is 2-dimensional, and the boundary of the region is a curve. Which of these has least length?

Figure 2

Questions of this type are quite difficult, even for regions in the boundary of a cube. If the sphere is replaced by the boundary of an arbitrary convex body, then it is impossible to expect precise results. But we might hope for an estimate.

It is interesting to note that the question we are posing has a different sort of answer for the sphere than for the ball. That is to say, consider these two questions:

• What is the region in the boundary of the sphere (in R³, let us say) that has perimeter curve of least length?
• What is the region inside the solid sphere (in R³) that has boundary surface of least area?

It turns out that the first question is answered by spherical caps cut off by hyperplanes. The second is answered by intersections of spheres. See Figure 3.

Figure 3

Let us now describe a philosophically related problem---one that is still wide open. Say that a convex body K in space is in isotropic position if

[1/vol(K)] ∫_K ⟨ x, y ⟩² dy = ||x||²

for every x in space. This condition just says that the body is not stretched more in any one direction than in any other. Any convex body may be put in isotropic position with a linear transformation of coordinates. Say that we have such a body, and it has volume equal to 1 unit. Is it true that any subset of measure 1/2 must have bounding area bounded below by some absolute constant? A related, perhaps easier (but still open) question is: Is it true that any such body (with unit volume) will have a hyperplane section with volume bounded below by some absolute constant? All of these problems are related to the issue of estimating the volumes of convex bodies. One of the key tools for that problem is sampling.

The first of these two questions comes from theoretical computer science, and arises from the issue of trying to produce a randomly distributed selection of points in the convex body. The only known method for producing such a selection is with a random walk. One would like to have estimates on the rate at which such a process gives a random distribution of points.

A classical result of Dyer, Frieze, and Kannan in 1991 is that the volume problem has complexity n²³ (where n is the dimension of the ambient space). Following a sequence of improvements in several papers, L. Lovasz and organizer S. Vempala in 2003 have reduced this estimate to n⁴. Recently (in 2006), Rademacher and Vempala showed that any algorithm that estimates the volume to a constant factor must have complexity at least n²/log n. This proof introduced important new tools in geometry. One would like to refine these estimates and have an exact idea of the complexity of the volume computation.

The subject of convex geometry has classical, geometrical roots. But today it interacts with theoretical computer science and probability. It is a dynamic area of discourse and research. The AIM workshop brought together workers from analysis, probability, algorithms, and statistics to compare techniques and develop collaborations.