如果你能躲在它后面，你能躲在它里面吗？
If you can hide behind it, can you hide inside it?
摘要：
设L是R^n中的紧凸集，且1 <= d <= n-1.定义集合L是d-可分解的，如果L是两个或多个维数不超过d的凸体的直接Minkowski和（仿射笛卡儿积）.称紧致凸集L为d-可靠的，如果L的每个d-维正交投影包含紧致凸集K的相应d-维投影的平移，则必有L包含K的平移.证明了当1 <= d <= n-1时：（1）d-可分解蕴含d-可靠性。(2)R^n中的紧致凸集L是d-可靠的当且仅当对所有m >= d+2，L的正则边界点没有m个单位法线形成（m-1）维单形的外单位法线。(3)光滑的凸体是不可靠的。(4)R^n中的紧致凸集L是1-可靠的当且仅当L是1-可分解的（即平行体）。(5)R^n中的中心对称紧凸集L是2-可靠的当且仅当L是2-可分解的。然而，存在非中心2-可靠凸体，它们不是2-可分解的。
英文摘要：
Let L be a compact convex set in R^n, and let 1 <= d <= n-1. The set L is defined to be d-decomposable if L is a direct Minkowski sum (affine Cartesian product) of two or more convex bodies each of dimension at most d. A compact convex set L is called d-reliable if, whenever each d-dimensional orthogonal projection of L contains a translate of the corresponding d-dimensional projection of a compact convex set K, it must follow that L contains a translate of K. It is shown that, for 1 <= d <= n-1: (1) d-decomposability implies d-reliability. (2) A compact convex set L in R^n is d-reliable if and only if, for all m >= d+2, no m unit normals to regular boundary points of L form the outer unit normals of a (m-1)-dimensional simplex. (3) Smooth convex bodies are not d-reliable. (4) A compact convex set L in R^n is 1-reliable if and only if L is 1-decomposable (i.e. a parallelotope). (5) A centrally symmetric compact convex set L in R^n is 2-reliable if and only if L is 2-decomposable. However, there are non-centered 2-reliable convex bodies that are not 2-decomposable.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
