We establish a new notion of tropical convexity for signed tropical numbers. We provide several equivalent descriptions involving balance relations and intersections of open halfspaces as well as the image of a union of polytopes over Puiseux series and hyperoperations. Along the way, we deduce a new Farkas' lemma and Fourier-Motzkin elimination without the non-negativity restriction on the variables. This leads to a Minkowski-Weyl theorem for polytopes over the signed tropical numbers.
@InProceedings{loho_et_al:LIPIcs.ITCS.2020.24, author = {Loho, Georg and V\'{e}gh, L\'{a}szl\'{o} A.}, title = {{Signed Tropical Convexity}}, booktitle = {11th Innovations in Theoretical Computer Science Conference (ITCS 2020)}, pages = {24:1--24:35}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-134-4}, ISSN = {1868-8969}, year = {2020}, volume = {151}, editor = {Vidick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.24}, URN = {urn:nbn:de:0030-drops-117097}, doi = {10.4230/LIPIcs.ITCS.2020.24}, annote = {Keywords: tropical convexity, signed tropical numbers, Farkas' lemma} }
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