凸体的同构截面 - 投影问题
The isomorphic section-projection problem for convex bodies
AI总结：
本文研究了凸体的同构截面-投影问题，证明了在特定条件下凸体体积的关系，并补充了同构Busemann-Petty和Shephard问题。
AI中文摘要：
设\(K,L\)为\(\mathbb{R}^n\)中的凸体，\(K\)为中心对称。假设对所有\(\theta\in S^{n - 1}\)，有\(|K\cap\theta^{\perp}|\leq|L\cap\theta^{\perp}|\)。我们证明\(|K|\leq c\sqrt{n}|L|\)，此结果在绝对常数选择上是精确的。该结果给出了混合截面 - 投影比较问题中的精确同构阶，补充了同构的布塞曼 - 佩蒂和谢泼德问题，还在绝对常数因子意义下消除了作者早期结果中的约翰位置假设。
英文摘要：
Let $K, L$ be convex bodies in $\mathbb{R}^n$ with $K$ centered. Assume that $|K \cap θ^{\perp}| \le |L|θ^{\perp}|$ for all $θ\in S^{n-1}$. We prove that $|K| \le c\sqrt{n}|L|$, which is sharp up to the choice of the absolute constant. The result gives the sharp isomorphic order in a mixed section-projection comparison problem, complementing the isomorphic Busemann-Petty and Shephard problems. It also removes the John's position assumption from an earlier result of the author, up to an absolute constant factor.
