Concepts of curvature in normed planes

V. Balestro, H. Martini and E. Shonoda, Concepts of curvature in normed planes, to appear in Expositiones Mathematicae, 2018. (arXiv link) (journal link)

Comments: This is an expository paper which aimed to gather together and study the concepts of curvature that one can define in a normed plane. There were three curvature types “spread” in the literature (even appearing in different papers and being defined in distinct ways), and our first step was to regard all of them as ways to measure the variation of the tangent vector of a curve. By adopting this point of view, we also define a new type of curvature. With the definitions in hand, we characterize curves with constant curvature, give formulas to obtain the curvature types with an auxiliary Euclidean structure, prove versions of the four vertex theorem for them, study closed curves and constant width curves, and define evolutes, involutes and parallels from distance squared functions.