Isoperimetric and isodiametric problems with constraints in Euclidean space

The isoperimetric and isodiametric problems in Euclidean space R n involve finding domains that maximize n-dimensional Lebesgue measure under specific constraints, such as the constant perimeter or diameter.In this paper, the following three aspects of the problems are investigated.(1) Strip-shaped domain: Consider a domain in R n of maximum measure, restricted in a strip-shaped region enclosed between two parallel hyperplanes, with constant diameter.(2) Cone-shaped domain: The aim is to find a domain of maximum measure with a constant diameter within an n-dimensional infinite right circular cone passing through its vertex.(3) Sector-shaped domain: A domain is searched in the plane with a minimal perimeter, within a sector-shaped region with a constant area.For each case, the existence and uniqueness of optimal domains are proven, their geometric shapes are determined, and their corresponding measures are calculated.