闭凸多胞体的调和中心：定义、计算及性质
Harmonic Center of a Closed Convex Polytope: Definition, Calculation and Some Properties
摘要：
考虑由m个线性不等式约束定义的n维闭凸多胞体。如果L是一条从任意可行点P向任意方向画出的直线，那么一般来说，它与每个约束相交于一点，得到m个交点。它表明，存在一个唯一的可行点Q沿着这条线的某处，使得1/di值的总和为0，其中di是Q和与约束i的交点之间的代数距离，沿着这条线测量。点Q被定义为沿着直线L的调和点。多面体的调和中心被定义为所有n条穿过它的直线的调和点，每条直线平行于一个坐标轴。证明了这种中心的存在性和唯一性。可以使用坐标搜索算法（CS）计算谐波中心，如一些简单的例子所示。这里定义的调和中心是前面定义的BI中心的推广，并且在几个方面更好。它表明，调和中心的多面体也是调和点的任何直线通过它在任何方向。证明了对于任意严格可行点P，存在唯一的调和超平面通过它，使得P是位于调和超平面内且通过P的任意直线的调和点。
英文摘要：
A closed convex polytope in n dimensions defined by m linear inequality constraints is considered. If L is a straight line drawn in any direction from any feasible point P, then in general, it intersects every constraint at one point, giving m intersections. It is shown that there exists a unique feasible point Q somewhere along this line, such that the sum of 1/di values is 0, where di is the algebraic distance between Q and the intersection with constraint i, measured along the line. The point Q is defined as the harmonic point along the line L. The harmonic center of the polytope is defined as that point which is the harmonic point for all n lines drawn through it, each parallel to one of the coordinate axes. The existence and uniqueness of such a center is shown. The harmonic center can be calculated using the coordinate search algorithm (CS), as illustrated with some simple examples. The harmonic center defined here is a generalization of the BI center defined earlier and is better in several respects. It is shown that the harmonic center of the polytope is also the harmonic point for any line drawn through it in any direction. It is also shown that for any strictly feasible point P, there exists a unique harmonic hyperplane passing through it, such that P is the harmonic point for any line which lies in the harmonic hyperplane and passes through P.
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。
