The ?-length Of Planar Convex Bodies And Isoperimetric Inequalities

The Brunn-Minkowski theory is the centre of convex geometry,then the Brunn-Minkowski inequality is the cornerstone of the classical Brunn-Minkowski theory.When the addition and multiplication was introduced into the theory,it developed to theL_p Brunn-Minkowski theory.It have been almost perfect,with the development of nearly twenty years of convex geometry.We notice that the L_p Brunn-Minkowski was studied when p31 and the p-order power is still the support function of the convex body that contains the origin in its interior.But when 0(27)p(27)1,the p-order power of the support function is not the support function.Because of the loss of convexity of the function,the original method of building the Brunn-Minkowski inequality and Minkowski inequality doesn't apply for the above.We chooseL_psurface as the studying object and try to explore the inequality of Brunn-Minkowski when 0(27)p(27)1.This article will study this problem on the two-dimensional plane.For this purpose,We define the?-perimeter of a convex body and propose the inequality of the Brunn Minkowski style.In this paper,we mainly study the Minkowski sum of the regular n-polygon and disc,proved the inequality holds in this situation.Based on this,if the inscribed circle of polygon and every side is tangent.The Minkowski sum of the n-polygon and disc with an inscribed also satisfies the inequality in the plane.