New results in
mathematics

To add two convex shapes, we take each point in one and add it to each point in the other. Collecting all these sums gives a new shape, called their Minkowski sum. How does the volume of the Minkowski sum relate to the volumes of the original shapes? Brunn proved a lower bound in 1887, and Minkowski showed in 1903 that if we scale the shapes before adding them then the volume of the sum depends polynomially on the scale factors. The polynomials that arise are called volume polynomials, and their coefficients describe the volumes and surface areas of the shapes along with new quantities that encode how the shapes interact. Inequalities between these coefficients are among the most important results in convex geometry, and volume polynomials have become central tools in algebraic geometry and combinatorics.

Which polynomials are volume polynomials? Heine answered this in 1938 for up to three shapes in the plane, and Shephard answered it in 1960 for two shapes in any dimension. The general case, of characterising the volume polynomials of any number of shapes in any dimension, is known as the Heine–Shephard problem and has been a central open problem in convex geometry.

In this paper, I resolve the Heine–Shephard problem in full generality. I give an explicit, enumerable family of quadratic inequalities with rational coefficients and show that a polynomial is a volume polynomial if and only if it satisfies all of them. I also show that, unlike every previously solved case, the space of volume polynomials for three or more shapes in three or more dimensions is not semialgebraic.

A polytope is a bounded shape with flat sides, generalising polygons to higher dimensions. One broad class of polytopes is the cubical polytope: a polytope is cubical if each of its sides is combinatorially a cube of one dimension lower, so that in three dimensions every side is a square, or in four dimensions every side is a cube. If we know how many vertices a cubical polytope has, what is the smallest number of edges and higher-dimensional faces it can have? The same number of vertices can be put together into many different cubical shapes, and in dimensions greater than three, some of these shapes use many more edges than others. One way we might reach the minimum is to start with some cube, and then glue further cubes one at a time, each onto a side of the shape built so far. This construction was described by Jockusch in 1993, whose conjecture is that all cubical polytopes formed in this way achieve the minimum.

In this paper, I prove Jockusch's conjecture in every dimension. The vertices of a cubical polytope can be coloured black and white so that every edge joins two colours, and in dimension four and above, I prove a stronger bound that grows with the difference between the numbers of black and white vertices.

An ellipsoid is a higher-dimensional generalisation of an ellipse. In \(n\) dimensions, we can make an ellipsoid by taking an \(n\)-dimensional sphere and stretching it by different amounts along each of the \(n\) axes, just as we make an ellipse by stretching a circle in the \(x\) and \(y\) axes. Up to rotation and translation, the ellipsoid is uniquely determined by these \(n\) stretch factors. But can we identify an ellipsoid using only the geometric properties we can measure, like volume or surface area? Convex geometry has a complete set of these measurements, called the intrinsic volumes of a shape. In this paper, I show that an ellipsoid is uniquely determined by its intrinsic volumes, in any dimension.

A lattice is a repeating arrangement of points in space, such as a square grid in two dimensions, or a cubic grid in three. Imagine placing an identical pulse of heat at every lattice point and letting it spread. Where are the coldest places on the grid, and do they move as time passes? This problem was first discussed by Baernstein in 1997. Shortly after the heat is released, we would expect the coldest places to lie in the middle of the largest gaps between lattice points. These are called the deep holes of the lattice. But as heat spreads further, more distant sources contribute and it is no longer clear that the deep holes remain the coldest. Baernstein showed that they always do for the hexagonal lattice in the plane, but for most other planar lattices the coldest places drift away from deep holes.

In this paper, I prove that for several highly symmetric lattices, including the eight-dimensional \(E_8\) lattice, which gives the densest packing of spheres in eight dimensions, the coldest points are always exactly the deep holes. I also prove that the same holds for several related lattices, including \(E_6\), the dual \(E_7^*\), the sixteen-dimensional Barnes–Wall lattice, and an infinite family of lattices in even dimensions.