On critical lattices of the unit sphere

The history of the problem of calculating and estimating the Hermite constant has two centuries. This article provides a brief overview of the history of this problem. Also, this problem is considered from the point of view of critical lattices of the unit sphere.
This problem begans from the works of J. L. Lagrange, L. A. Seeber and K. F. Gauss. While developing the theory of reduction of positive definite quadratic forms, they obtained limit forms for which the ratio of the minimum value of these forms at integer points other than 0 to their determinant is maximal.
In the middle of the 19th century, Sh. Hermit obtained an estimate for this quantity for an arbitrary dimension. And at the end of the 19th century, A. N. Korkin and E. I. Zolotarev proposed a new method for reducing quadratic forms, which made it possible to obtain exact values of the Hermite constant up to dimension 8.
In this paper, we will consider a quantity equivalent to the Hermite constant, the critical determinant of the unit sphere. It should be noted that these quantities are closely connected with other problems in the geometry of numbers, for example, the problems of finding the density of the best packing, finding the shortest lattice vector, and Diophantine approximations. We present critical lattices of dimensions up to 8 and consider some of their metric properties.