Separation Theorems in Signed Tropical Convexities

Explore separation theorems in signed tropical convexities through this 32-minute lecture by Mateusz Skomora from the Hausdorff Center for Mathematics. Delve into the world of max-plus semifields and tropical convexity, comparing it to standard convexity over nonnegative real numbers. Discover the limitations of tropical polyhedra and learn about the TO-convexity and TC-convexity as solutions to these constraints. Examine the structural results of TC-convexity, including the hyperplane separation theorem, characterization of signed tropical segments and hemi-spaces, and analogues of the Carathéodory and Minkowski-Weyl theorems. Gain insights into the relationship between TC-convex and TO-convex hulls for generic point collections. The lecture covers an introduction, TO-convexity definition, segments in TO-convexity, halfspaces, left sum of signed numbers, TC-convexity properties, Carathéodory's theorem, separation in TC-convexity, and concludes with open problems and areas for further study.

Intro
Outline
TO-convexity definition
Segments in the TO-convexity: generic case
TO-convexity and halfspaces
Left sum of signed numbers
Segments in the TC-convexity
First properties of TC-convexity
Carathéodory's theorem
Separation in the TC-convexity
Open problems and further study