A function is said to be differentiable at if there exists such that We say that is the gradient of at and write

Define the Voronoi cell of a lattice

let be a continuously differentiable function, and put

for an invertible matrix . Then is differentiable at the identify matrix and

Let be a symmetric convex body. We can define a norm We can define the matrix norm on where .

The polar body of a symmetric convex body is defined as

For any symmetric convex body there exists such that

Let be the space of matrices with zero trace. The Laplacian of a twice differentiable function is defined by directional derivative is define by