A New Method To Study The Containment Measure Of Convex Bodies In Light Of Minkowski Subtractions

Convex bodies and star bodies are the main research objects in the paper. The paper is concerned with three aspects: the integral formula for the containment measure of a line segment inside a skew cylinder and its application to the Buffon-Ren problem; a new method to study the containment measure of a compact convex set in a convex body in light of Minkowski subtractions; the multivariant abstractions of some geometric inequalities of star bodies.1. The integral formula for the containment measure of a line segment inside a skew cylinder and its application to the Buffon-Ren problem.This part belongs to integral geometry. In light of basic theory of Euclidean integral geometry, the author obtains the integral formula for the containment measure of a line segment inside a skew cylinder and applies this result to solve the Buffon-Ren problem with parallelotopes as basic region.2. A new method to study the containment measure of a compact convex set inside a convex body in light of Minkowski subtractions.This part belongs to theoretical study of the containment measure in integral geometry and includes the following results.(1) Based on the basic theories of integral geometry and convex geometry, this method is introduced and the fundamentals are given.(2) As examples, this method yields some kinematic formulas of containment measure, and the extreme problem for the translation containment measure is solved.(3) Applied to study the containment measure of a line segment inside a convex body, this method yields some new integral formulas, and the integral formula for containment measure of a line segment inside a ellipsoid in higher Euclidean space are obtained.(4) In light of convex geometry, the details for this method is studied, and some formulas for containment measure of compact convex sets inside convex bodies are obtained.(5) As an application, a global property about the translation containment measure of simplexes with same dimension inside a convex body is shown. This global property is the generalization of the relationship between the containment measure of lines in a convex body and the chord power integral of a convex body.3. The multivariant abstractions of some geometric inequalities of star bodies.This part belongs to the dual Brunn-Minkowski theory. First, the following notions are introduced: radial multiplicative operations of star bodies; radial power operations of star bodies; multivariant dual mixed volumes of star bodies. Second, based on such notions, the multivariant version form of dual Minkowski inequality and abstraction versions of some geometric inequalities of star bodies are obtained. Finally, the radial average body of star bodies is studied and some applications are shown.