The structure of max-plus hyperplanes
The authors study max-plus hyperplanes in the semimodule \(R^n_{\max}\) over the max-plus semifield \(R_{\max}\), where \(R_{\max}:= R\cup\{-\infty\}\) is with the operations \(\oplus:= \max\) and \(\otimes:= +\). From the authors' summary: We give a formula that expresses a nondegenerate strictly affine hyperplane as a union of complements of semispaces at a point \(z \in R^n\), called the ``center'' of \(H\), with the boundary of a union of complements of other semispaces at \(z\). Using this formula, we obtain characterizations of nondegenerate strictly affine hyperplanes with empty interior. We give a description of the boundary of a nondegenerate strictly affine hyperplane with the aid of complements of semispaces at its center, and we characterize the cases in which the boundary \(bd (H)\) of a nondegenerate strictly affine hyperplane \(H\) is also a hyperplane. Next, we give the relations between nondegenerate strictly affine hyperplanes \(H\), their centers \(z\), and their coefficients. In the converse direction we show that any union of complements of semispaces at a point \(z \in R^n\) with the boundary of any union of complements of some other semispaces at that point \(z\), is a nondegenerate strictly affine hyperplane. We obtain a formula for the total number of strictly affine hyperplanes. We give complete lists of all strictly affine hypeprlanes for the cases \(n=1\) and \(n=2\). We show that each linear hyperplane \(H\) in \(R_{\max}^n\) can be decomposed as the union of four parts, where each part is easy to describe in terms of complements of semispaces, some of them in a lower dimensional space.
scientific article; zbMATH DE number 5190949