The Mean Length Of The Chords Of A Convex Domain

As an independent branch of mathematics, integral geometry and geometric probability is considered to have been formally established in the 1930, s . It has accumulated academic harvests in abundance and wide applications in many fields. The average length of the chords inside the convex area is one of the most important problems, it not only has academic meaning but has a application in the architecture acoustics.We can have reference to the works of Spain mathematician L.A.Santalo whose name was《Integral Geometry and Geometric Probability》,it gave the definition method to get the average length of the chords inside the bounded convex body K ,but not give the general idea to concrete the average length of the chords inside the convex area.This paper mainly on the above-mentioned foundation that have already had the average length definition of the chords inside the convex area carries on transformation, namely changes into E₁(σ)= I₂/Ï€F (2)according to the definition of the integral for the power of chords inside the bounded convex body K, will be recorded for I₂, then we can make use the generalized support function to get I₂, finally we take it into the formula(2) and concretely get the average length of the chords inside the convex area requested.This thesis gives the concrete calculation result of the circle area ,the rectangle area and the ellipse area; Besides,we can calculate the maximum value of the the mean length inside the rectangle area and draw a comparison between the ellips area and the circle area,then we can get some useful conclusions.The ideas of this thesis have wide applications in concreting the average length of the chords inside the convex area.They are available in any plane convex areas.