Some Properties Of The Critical Set Of The Dual Mean Minksowski Measure Of Symmetry

In this thesis,we focus on the properties and the related problems of the critical set of the dual mean Minkowski measure of symmetry,a kind of geometric affine invariants for convex bodies proposed by Q.Guo.In order to figure out the relation between the dual mean Minkowski measures of symmetry and classical Minkowski one,we present a formula for calculating the value of the dual mean Minkowski measure at a Minkowski critical point.For getting such a formula,we establish a theorem of Helly's type,both in analytic form and in geometric form,on the families of(partially closed)half-spaces.At the end,we show that if a convex body has a regular point with respect to the dual Minkowski measures,then there are n+1 affine diameters passing through this point,which in part answers positively a conjecture proposed by Grünbaum: each convex body has n+1affine diameters passing through a fixed point.Main results obtained in this thesis are as following:(1).Theorems of Helly's type on the families of partially-closed half-spaces in analytic and in geometric forms;(2).Formula for calculating the value of the dual mean Minkowski symmetry at a Minkowski-critical set;(3).A partial positive answer to the Grünbaum Conjecture.