给定拓扑的最小树的长度：Maxwell公式的推广
The Length of a Minimal Tree With a Given Topology: generalization of Maxwell Formula
摘要：
经典的Maxwell公式根据边界顶点的坐标和引入边的方向计算平面局部极小二叉树的长度。然而，如果具有给定拓扑和边界的极端树具有退化边，则经典的麦克斯韦公式不能直接应用，在这种情况下，为了计算极端树的长度，必须知道哪些边是退化的。本文将Maxwell公式推广到任意维数的欧几里得空间中的任意极值树。现在要计算这样一棵树的长度，既不需要知道哪些边是退化的，也不需要知道非退化边界边的方向。其结果是某个特殊线性函数在欧几里得空间中与某些圆柱相交的紧凸子集上的最大值。
英文摘要：
The classic Maxwell formula calculates the length of a planar locally minimal binary tree in terms of coordinates of its boundary vertices and directions of incoming edges. However, if an extreme tree with a given topology and a boundary has degenerate edges, then the classic Maxwell formula cannot be applied directly, to calculate the length of the extreme tree in this case it is necessary to know which edges are degenerate. In this paper we generalize the Maxwell formula to arbitrary extreme trees in a Euclidean space of arbitrary dimension. Now to calculate the length of such a tree, there is no need to know either what edges are degenerate, or the directions of nondegenerate boundary edges. The answer is the maximum of some special linear function on the corresponding compact convex subset of the Euclidean space coinciding with the intersection of some cylinders.
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