A Generalization of the Convex Kakeya Problem

Abstract

Given a set of line segments in the plane, not necessarily finite, what is a convex region of smallest area that contains a translate of each input segment? This question can be seen as a generalization of Kakeya's problem of finding a convex region of smallest area such that a needle can be rotated through 360 degrees within this region. We show that there is always an optimal region that is a triangle, and we give an optimal Θ(n log n)-time algorithm to compute such a triangle for a given set of n segments. We also show that, if the goal is to minimize the perimeter of the region instead of its area, then placing the segments with their midpoint at the origin and taking their convex hull results in an optimal solution. Finally, we show that for any compact convex figure G, the smallest enclosing disk of G is a smallest-perimeter region containing a translate of every rotated copy of G.

Photo Composition with Real-Time Rating

“…An example is shown in Figure 10. As discussed previously in the rule of thirds case, we can find the salient region of the main object and then use Ahn et al [24] proposed method to find the smallest triangle to enclose the salient region. For example, as shown in Figure 11, the red triangle is the smallest triangle, including the largest salient region of Figure 7f.…”