热带凸性在选址问题中的应用
Tropical convexity in location problems
AI总结：
本文研究了在热带凸包中求解选址问题的方法，证明了在特定距离度量下最优解的性质，并探讨了其在系统发育学中的应用。
摘要：
我们研究了最优解位于输入点的热带凸包中的选址问题。首先，我们研究了在非对称热带距离下的测地星形集，并引入了热带准凸函数的类别，其子水平集具有这种形状，与单调函数相关。然后我们证明，当距离由之前提到的热带准凸函数测量时，选址问题的最优解位于输入点的热带凸包中。我们还证明，如果将输入点替换为热带凸集，也会得到类似的结果。最后，我们专注于系统发育学中的应用，展示了从我们这一类选址问题中出现的共识方法的性质。
英文摘要：
We investigate location problems whose optimum lies in the tropical convex hull of the input points. Firstly, we study geodesically star-convex sets under the asymmetric tropical distance and introduce the class of tropically quasiconvex functions whose sub-level sets have this shape. The latter are related to monotonic functions. Then we show that location problems whose distances are measured by tropically quasiconvex functions as before give an optimum in the tropical convex hull of the input points. We also show that a similar result holds if we replace the input points by tropically convex sets. Finally, we focus on applications to phylogenetics presenting properties of consensus methods arising from our class of location problems.
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