同时的Busemann-Petty与Shephard体积比较
Simultaneous Busemann-Petty and Shephard Volume Comparisons
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中文总结 AI 辅助
该研究针对不同维度构造凸体反例，探讨中心超平面截面与正交超平面投影的体积比较能否确定两凸体体积的大小顺序。
中文摘要 AI 辅助
我们研究同时的Busemann-Petty与Shephard体积比较问题：所有中心超平面截面的体积和所有正交超平面投影的体积比较，是否能确定两个凸体体积的大小顺序。对每个n≥5，我们构造了ℝⁿ中原点对称的旋转凸体K、L，使得K的每个中心超平面截面和每个正交超平面投影的体积都严格小于L对应的截面或投影，而|K|>|L|。对n≤4，Busemann-Petty问题的肯定解表明仅截面不等式就意味着|K|≤|L|。在非原点对称的情况下，我们在每个维度n≥2中构造了此类反例，其中一个物体是欧氏球的非平凡平移，另一个是常亮度的非中心对称物体。
英文摘要
We study the simultaneous Busemann-Petty and Shephard volume comparison problem: whether comparison of the volumes of all central hyperplane sections and all orthogonal hyperplane projections determines the ordering of the volumes of two convex bodies. For every $n\geq5$, we construct origin-symmetric convex bodies of revolution $K,L\subset {\mathbb R}^n$ such that every central hyperplane section and every orthogonal hyperplane projection of $K$ has strictly smaller volume than the corresponding section or projection of $L$, while $|K|>|L|$. For $n\leq4$, the affirmative solution of the Busemann--Petty problem shows that the section inequalities alone imply $|K|\leq|L|$. Without origin symmetry, we construct such counterexamples in every dimension $n\geq2$, with one body a nontrivial translate of a Euclidean ball and the other a noncentrally symmetric body of constant brightness.
