“a week ago i chatted with @AchimWar about favourite problems and the caratheodory…”

a week ago i chatted with @AchimWar about favourite problems and the caratheodory conjecture came up. i literally wanted to just read about it but i did see the complicated literature situation and so in hopes i could help asked him if it was ok to look at with a model. me, John-Paul Smith, and of course claude checked and (i hope im being a better citizen, hadamard led to a lot of flaming) i sent over a pdf writeup, and for the past week it’s circulated round. i did wanna have it recorded somewhere public though so i’ll try to again give a form certifiable by models

C^\infty caratheodory and loewner are false, let f(x + iy) := -\cos(2x) / 4 + 3\cos(2y) / 10 - \cos(4y) / 32 + \sin(x)\sin(y), and, for k\in \Z^+, g_k(z) := |z|^2 e^{-|z|^{-1/4} e^{-|z|^2}} f((100 / \bar{z})^{k/2}) / (1 + |z|^2) + 10^{10}. then g_k has an index 1 + k/2 umbilic point at 0, and g_2, regarded as C^\infty on S^2 = \P^1(\C) thanks to the e^{-|z|^2}, is the support function of a convex body with exactly one umbilic point.