凸体内点与边界点之间的平均距离
Mean distance between points inside and on the boundary of a convex body
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中文总结 AI 辅助
本文研究扎波罗热茨-塔拉索夫猜想，针对中心对称凸体、平面凸体、外切凸体等不同情况，通过构造反例、推导公式等方式取得相关结论，还证明该猜想对三角形的逐投影形式成立。
中文摘要 AI 辅助
本文研究扎波罗热茨-塔拉索夫猜想，该猜想指出凸体内两个随机点之间的平均距离不超过其边界上两个随机点之间的平均距离。对于中心对称凸体，该问题已完全解决：平面凸体的猜想成立，而所有三维及以上维度的凸体中该猜想均不成立，且对距离的所有矩均如此；同时构造了显式反例族及维度提升构造。还证明对于足够高的矩，任意平面凸体都存在类似不等式。对于外切凸体，得到了平均距离间的精确关系，即通过本文引入的投影分布推导得出的金曼公式的类似形式；此外，该猜想对三角形以更强的逐投影形式成立。
英文摘要
The Zaporozhets--Tarasov conjecture is considered, which states that the mean distance between two random points inside a convex body does not exceed the mean distance between two random points on its boundary. For centrally symmetric bodies the question is settled completely: the conjecture is proved for planar bodies and disproved in every dimension at least three, for all moments of the distance; an explicit family of counterexamples is constructed together with a dimension-lifting construction. It is also shown that for sufficiently high moments an analogous inequality holds for any planar convex body. For circumscribed bodies exact relations between the mean distances are obtained --- analogues of Kingman's formula derived via the projective distributions introduced here; moreover, the conjecture is proved for triangles in a stronger per-projection form.
