The Existence Of The Positions Of Minimal Surface Areas Of A Convex Body

In the 1970s,Petty introduced minimal surface area to convex bodies,and strengthened the classical isoperimetric inequality into affine isoperimetric classic isoperimetric inequality by minimal surface area,and initially established the relationship between affine isoperimetric inequality and measure isotropy.Petty’s work inspired a series of important research works such as ball’s volume ratio problem of John’s ellipsoid and the reverse isoperimetric problem of the minimum surface area,Lutwak-Yang-Zhang’s reverse affine isoperimetric problem of Lp John’s ellipsoid and Lp minimum surface area,Zou-Xiong’s research on Orlicz John’s ellipsoid and related affine extremum problems.The research content of this paper belongs to Brunn-Minkowski theory.It mainly studies the existence of the position of minimum surface area of convex body.The surface area of each order S1,...,Sn-1 of convex body is rigid invariant,but not affine invariant.In order to extract affine invariants,the following affine geometry systems are studied Aj(K)=inf {Sj(gK):g∈SL(n)} j=1,...,n-1.For this case j=n-1,Petty has proved the unique existence of the above-mentioned affine extremum problem,and established the demonstrative theorem of the solution.Affine geometric quantity An-1(K)is exactly the minimum surface area of Petty.In 2000,the famous mathematicians Milman and Giannopoulos advocated to study the above-mentioned affine geometric quality system,and for this new situation j<n-1,under the premise of assuming that there is a solution,the necessary conditions for the solution of the affine extremum problem are established.However,for 20 years,the existence of solutions has been in the air.The main work of this paper is to proved the existence of the solution for the affine extremum problem j<n-1.In the past,the research scheme of situation j-n-1 did not use this low-dimensional situation j<n-1.In this paper,the integral geometry and convex geometry are used to solved the problem.