Research On The Dual Chernoff Type Inequality And Its Stability

The Chernoff type inequality is one of the important geometric in-equalities.This paper mainly takes planar star bodies as the reseach object,and obtains the extension and stability of the dual Chernoff type inequalities by using Fouries series.In addition,we also discuss a class of construction methods of com-plete sets in Banach space.The details are as follows:(1)For planar star bodies,the dual Chernoff-Ou-Pan inequality is generalized,which based on the dual Chernoff-Ou-Pan inequality of Zhang-Yang,and thus the generalized form of the dual Chernoff-Ou-Pan inequality is established.At the same time,another proof of the classial dual isoperimetric inequality is given.Refer to the second section of the third chapter in this dissertation.(2)Inspired by the mixed symmetric Chernoff type inequality of Zhang,we consider the mixed Chernoff inequality for two star bodies,and the dual mixed sym-metric Chernoff type inequality for two planar star bodies is studied.Furthermore,the stability estimation of this inequality is obtained by using the dualL₂metric.Refer to the first section and the second section of the fourth chapter.(3)The constructions of complete sets in Banach space are investigated.For the constructions of complete sets,Bavaud,Lachand-Robert and Oudet constructed com-plete sets inn+1 dimension Euclidean space from the complete sets inndimension Euclidean space.Papini and Wu extended the results of Bavaud's,Lachand-Robert and Oudet's to Banach space.Based on the Papini and Wu's work,the construc-tions methods of Bavaud's,Lachand-Robert and Oudet's are futher extended to high dimensional Banach space by replacing hyperplane with arbitrary subspace,that is,from the complete sets in lower dimensional Banach space to construct the complete sets in the arbitrary finite dimensional Banach space.Refer to the fifth chapter.