Problems Related To The Position Of Minimum Surface Area Of Convex Bodies

This paper belongs to Brunn-Minkowski theory and mainly studies the problem of extremum location associated with convex surface area.In 1961,by solving the affine extremum problem of convex bodies.Petty introduced minimal surface area into convex bodies.From minimal surface area,the classical isoperimetric inequality was strengthened to affine isoperimetric inequality,and the relationship between affine isoperimetric inequality and isotropy of convex surface area measure is established.The surface areas of convex bodies are rigid invariant,but not affine invariant.In order to extract affine invariants from rigid body invariants of convex surface area.In this paper,we study the following affine geometric quantities:A_j(K)=inf{S_j(gK):g?SL(n)},j=1,…,n-1.When j=n-1,Petty has proved the expression of the minimum surface area of a convex body in affine geometric quantity problem,and established an indication theorem for understanding,where the affine geometric quantity An-1(K)is the surface area studied by Petty.The main work of this paper is to solve the above affine minimization problem in the case of j<n-1.It is proved that under the volume preserving affine image,the transformation g satisfies some conditions to make the convex body have the minimum surface area,and solve the problem of minimizing the surface area of the convex body.Furthermore,an indication theorem is established for the position of the minimum surface area of a convex body and the relationship between the extremum body at the position of the minimum surface area and the isotropic measure of a convex body is revealed.