Given a set of $n$ colored points with $k$ colors in the plane, we study the problem of computing a maximum-width rainbow-bisecting empty annulus (of objects specifically axis-parallel square, axis-parallel rectangle and circle) problem. We call a region rainbow if it contains at least one point of each color. The maximum-width rainbow-bisecting empty annulus problem asks to find an annulus $A$ of a particular shape with maximum possible width such that $A$ does not contain any input points and it bisects the input point set into two parts, each of which is a rainbow. We compute a maximum-width rainbow-bisecting empty axis-parallel square, axis-parallel rectangular and circular annulus in $O(n^3)$ time using $O(n)$ space, in $O(k^2n^2\log n)$ time using $O(n\log n)$ space and in $O(n^3)$ time using $O(n^2)$ space respectively.

翻译：给定平面上一组具有 $k$ 种颜色的 $n$ 个彩色点，我们研究计算最大宽度彩虹平分空心环（具体对象为轴平行正方形、轴平行矩形和圆形）的问题。若一个区域包含每种颜色的至少一个点，则称该区域为彩虹区域。最大宽度彩虹平分空心环问题要求寻找一个特定形状的环 $A$，使其宽度尽可能大，且 $A$ 不包含任何输入点，并将输入点集平分为两个部分，每个部分均为彩虹。我们通过 $O(n^3)$ 时间和 $O(n)$ 空间计算最大宽度彩虹平分空心轴平行正方形环，通过 $O(k^2n^2\log n)$ 时间和 $O(n\log n)$ 空间计算最大宽度彩虹平分空心轴平行矩形环，并通过 $O(n^3)$ 时间和 $O(n^2)$ 空间计算最大宽度彩虹平分空心圆形环。