| The chord length distribution function of convex domain is one of the most inportant subject of convex body theory. It has much application background such as Pattern Recognition,Statistical analysis of materials.But now ,there is not provide a unified approach to obtain the chord length distribution function for convex domain in existing literature.In this paper we take rectangle for example to discuss the approach to calculate the exact analytical formula of chord length distribution function in using generalization support function technique and limited chord function. This method can also be used in convex domain with parallel sides.Classic Brunn-Minkowsi inequality has the profound connotation of the geometry, which is the Brunn-Minkowsi theory,The Brunn-Minkowsi theory links Minkowsi sum with volume closely,which can be applied to allareas of mathematics,and becoming a powerfull tool to slove the difficult problem of surface area, volume, width and othermetric relations. The origins of Lp-Brunn-Minkowski is Firey linear combination of convex body,which was difined by Firey in 1962. We owe the theory of relativity to famous mathematician E.Lutwak, so that Firey Lp combination is introduced to the classical Brunn-Minkowsi thery,and he put ofwward concept of Lp mixing volume,Lp mixing homogeneous integral,Lp surface area measuremnet and Lp-mixed surface area,and established the corresponding integral equation.Science then,the classical Brunn-Minkowsi theory extension to Lp space . As the core of inequalities, Brunn-Minkowsi inequality connects a series of related affine isoperimetric inequalities,such as Petty projective inequality and affine Sobolev inequality and Lp affine Sobolev inequality . These classical inqualities have very stong aptitude . Based on the Brunn-Minkowsi theory and Lp-Brunn-Minkowski theory,this paper presents the proof of equivalence between Lp projective inequality and Lp centroid inequality. |
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