On a cosine function defined for smooth normed planes

V. Balestro and E. Shonoda, On a cosine function defined for smooth normed planes, Journal of Convex Analysis 25(1), pp. 21-39, 2018. (arXiv link) (journal link)

Comments: We study a cosine function that can be defined for smooth normed spaces (in the sense that each point of the unit sphere has a unique supporting hyperplane). We derive a geometric flavored definition for it, and relate it to the Minkowski sine function. As applications, we give an expression to the Gateaux derivative of the norm, derive a local characterization of Radon planes (which was still missing, as far as I know), and obtain a formula to measure the distortion between the lengths of external tangents drawn from a point to the unit circle of a Radon plane. A new geometric characterization of Radon curves is also derived.