Geometric inequality for pedal simplex and its applications

The n-dimension simplex is the basic convex body in the n-dimension Euclidean space.Its geometric nature is usually universal.Recently,many important geometric inequalities have been established for the geometric inequality of n-dimensional simplex.However,the study concerning the geometric inequality problem for the pedal simplex is very less.The inequalities for volumes of a simplex and its pedal simplex were investigated.Using analytic method and theory of geometric inequality,the problem of geometric inequality for the pedal simplex of a simplex in En was discussed.An inequality for the circumradius and inradius of a simplex and its pedal simplex was established.As the special case,some generalizations of the famous n-dimensional Euler inequality were obtained.