Critical Determinant

In this chapter, we introduce the concept of critical determinant of a set in \(\mathbb {R}^n\) and discuss its arithmetical significance. We define the notion of critical lattice of a set and investigate its existence. A critical lattice of a bounded set has no point other than origin O in its interior, but it can have several points on its boundary. We investigate maximum and minimum number of points a critical lattice of a convex body \(\mathcal {K}\) can have on boundary of \(\mathcal {K}\) . We obtain critical determinants and critical lattices of convex domains with center O in \(\mathbb {R}^{2}\) . It is also shown that critical determinant of a symmetrical convex domain \(\mathcal {K}\) with center O is equal to one fourth of its area if and only if \(\mathcal {K}\) is either a parallelogram or a symmetrical hexagon with center O. We also determine critical determinant and all critical lattices of a sphere in \(\mathbb {R}^{3}\) and give some historical remarks.