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 ≤ lnN-ln（N-1）/ln2＋2[lnN-ln（N-1）], then （1/N N↑∑↑i=1di^r, k）^1/r≤1/N coskπ/N N↑∑↑i=1｜AiAi＋k｜≤sin2kπ/2sinπ/N（1/N N↑∑↑i=1｜AiAi＋1｜^2.
基金
Supported by the NSF of China(10171073)
