子空间维数为2和3的广义Busemann-Petty问题的正解
An affirmative solution to the generalized Busemann--Petty problem with subspace dimensions $2$ and $3$
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中文总结 AI 辅助
本文解决了n≥5时m=2、3的广义Busemann-Petty问题低维遗留情形，结合已有结果完成了该问题的完全解答。
中文摘要 AI 辅助
广义Busemann-Petty问题问：若K和L是Rⁿ中关于原点对称的凸体，对每个m维子空间E（1<m<n），K∩E的m维体积不大于L∩E的m维体积，是否能推出K的n维体积≤L的n维体积？超平面情形m=n-1是经典Busemann-Petty问题，n=3、4时答案为正，n≥5时为负。广义问题中，当3<m<n时答案为负，而n≥5时m=2、3的低维情形自1990年代起悬而未决，本文解决了这些遗留情形，结合已有结果，广义Busemann-Petty问题得到完全解决。
英文摘要
The generalized Busemann--Petty problem asks whether origin-symmetric convex bodies in $\mathbb{R}^n$ having larger volume of all $m$-dimensional sections necessarily have larger volume. When $m\geq 4$, this is known to be false, but the cases $m=2, 3$ for $n\geq 5$ have remained open since the 1990s. In this paper, we resolve these cases. Together with the known results, the generalized Busemann--Petty problem is completely solved: the answer is affirmative for $m=1, 2, 3$, and negative for $m\geq 4$.
