Shapes with offsets of nearly constant surface area

This article addresses several issues of offset sizes. A new type of isoperimetric problems is introduced: find the shapes, which have offsets of minimal surface area change. This problem arises in some physical and chemical processes with constant energy release. A computational solution is presented for two cases: convex sets, and a subclass of non-convex sets (star-like sets). Both cases are discussed in the plane and in the space. The solution uses methods from convex theory including the theory of mixed volumes and also some optimization techniques. The formulas developed in the optimization problem can be applied to get analytical formulas of the length of the curve offsets, as well as of the surface area and the volume of the surface offsets. Evaluating properties of offsets without constructing them proves useful for preliminary design in solid modeling, for approximating offsets curves and for planning the velocity of offset paths.