AI 中文总结
该数学研究证明任意平面凸体内部两点平均距离严格小于边界上两点平均距离，通过基尼均值差一维比较等方法完成，相关验证材料含6492个系数。
AI 中文摘要
我们证明：在任意平面凸体中，两个独立均匀分布点之间的平均距离严格小于其边界上两个独立均匀分布点之间的平均距离。两者不可能相等，不过沿一系列薄矩形，这两个平均距离的差值趋于零。该证明将问题简化为两个基尼均值差的一维比较。主要新要素是关于随机对(ε,R)∈{-1,1}×[0,1]的矩引理。证明中唯一的有限代数部分由伯恩斯坦基中的精确有理证书给出；验证脚本及全部6492个系数均见于补充材料。
英文摘要
We prove that the mean distance between two independent uniformly distributed points in an arbitrary planar convex body is strictly smaller than the mean distance between two independent uniformly distributed points on its boundary. Equality is impossible, although the difference between these two mean distances tends to zero along a sequence of thin rectangles. The proof reduces the problem to a one-dimensional comparison of two Gini mean differences. The main new ingredient is a moment lemma for a random pair $(\varepsilon,R)\in\{-1,1\}\times[0,1]$. The only finite algebraic part of the proof is given by exact rational certificates in the Bernstein basis; the verification script and all 6492 coefficients are available in the supplementary materials.
Comments35 pages, 3 figures. Russian manuscript completed and privately deposited on GitHub on July 26, 2026: https://github.com/mkukushkin2004/mean-distance-inequality-proof/commit/17c5b853b52146f7960f286565a1492fb334424c. Repository public since August 13, 2026. English translation; mathematical text unchanged except for an Author's note
