Disjoint Placement Probability of Line Segments via Geometry

摘要

© 2023 The Mathematical Association of America.We have shown that when any finite number n, of line segments with total combined length less than one, have their centers placed randomly inside the unit interval (Formula presented.), the probability of obtaining a mutually disjoint placement of the segments within (Formula presented.), is given by the expression (Formula presented.) where (Formula presented.), and (Formula presented.) denotes the length of the k-th segment, Lk. The result is established by a careful analysis of the geometry of the event, “all segments disjoint and contained within [0,1],” considered as a subset of the uniform probability space of n centers, each of which is in (Formula presented.); that is to say, the unit n-cube of (Formula presented.). This event has an interesting geometric structure consisting of (Formula presented.) disjoint, congruent, (up to a mirror image) polytopes within the unit n-cube. It is shown these event polytopes fit together perfectly to form, except for a set of measure zero, a partition of an n-dimensional cube with common edge length (Formula presented.), and hence an n-volume given by the formula. In the case of n = 3 segments, the polytopes form one of the known tetrahedral partitions of the cube as discussed, for example in [4]. In fact for all n > 0, the polytopes comprise a partition of the n-dimensional hypercube, and are therefore n-dimensional space filling.