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The local Steiner problem in normed planes

We present a geometric analysis of the local structure of vertices in a Steiner minimum tree in an arbitrary normed plane in terms of so-called absorbing and critical angles, thereby unifying various results known for specific norms. We find necessary and sufficient conditions for a set of segments emanating from a point to be the neighborhood of a vertex in a Steiner minimum tree. As corollaries, we show that the maximum possible degree of a Steiner point and of a given point are equal, and equal 3 or 4, except if the unit ball is an affine regular hexagon, where it is known that the maximum degree of a Steiner point is 4 and of a regular point is 6. We also characterize the planes where the maximum degree is 4, the so-called X-planes, and present examples. In particular, if the unit ball is an affine regular 2n-gon, Steiner points of degree 4 exist if and only if n = 2, 3, 4, or 6.