The geometry of one-dimensional and spatial Hardy type inequalities

The proofs of many hardy-type inequalities are based on one-dimensional inequalities. The difficulties that come from the domains of integration are implicitly reflected in the one-dimensional inequalities on the interval used to substantiate the spatial analogs. One-dimensional inequalities are the analytical basis for solving geometric problems. The paper provides a brief overview of the results in this direction. An attempt is made to systematically present the theory of Hardy-type inequalities with additional terms involving the geometric characteristics of the regions, for example, such as the volume, diameter, inner radius, or the maximum conformal modulus of the region.