关于多边形面积的Oppenheim不等式的推广(英文)

A GENERALIZATION OF OPPENHEIM'S AREA INEQUALITY FOR POLYGONS

设a_1,a_2,…,a_n,F_1和b_1,b_2,…,b_n,F_2是两个n边形的边长和面积。若c_i=(a_i^P+b_i^P)^(1/P),i=1,2,…,n,且p≥1,则当2≤p≤4时,以c_1,c_2,…,c_n为边长的n边形的最大面积满足:F^(P/2)≥F_1^(P/2)+F_2^(P/2)。这不等式对p>4不成立,但当1≤p<2时仍悬而未决。

Let a_1, a_2, …, a_n, F_1, and b_1, b_2, …, b_n, F_2 be the sides and areas of two n-gons. If c_i=(a_i^P+b_i^P)^(1/P), i=1, 2,…, n, and p≥1, then c_1, c_2,…, c_n are the sides of a n-gon, and the maximum areas satisfy F^(F/2)≥F_1^(P/2)+F_2^(P/2) for 2≤p≤4. The inequality does not hold for p>4. It is not known whether it holds for 1≤p<2.