含k-Brocard距离的一类几何不等式

摘要

Let P be an inner point of a convex N-gon ΓN: A1A2 …ANA1(N≥3), and let di,k denote the distance from the point Ai+k to the line PAi(i=1,2,… ,N, Ai=Aj(→)i≡j(modN)), which is called the k-Brocard distance for P of ΓN. We have proved the following double-inequality: If P ∈ΓN, k = N∩i=1∠Ai-kAiAi+k(1≤k＜N/2,i=1,2,…,N),and r≤1nN-1n(N-1)/1n2+2[1nN-1n(N-1)],then(1/NN∑i=1dri,k)1/r≤1/Ncoskπ/NN∑i=1|AiAi+k|≤sin2kπ/N/2sinπ/N(1/NN∑i=1|AiAi+1|2)1/2