As of: 2026-10-30 13:42 CET.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/p/x-__alpoge__-2089971359921156203/

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

levent (@__alpoge__), X, 2026-08-19.
Source: https://x.com/__alpoge__/status/2089971359921156203

## Archived text

> 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.

Archived 2026-10-28 via fxtwitter (unofficial).
Counts: 107,238 views, 461 likes, 42 reposts, 27 replies (at fetch time)

## Cited in

- 2026-08-19: [Smooth Carathéodory and Loewner conjectures disproved with Claude](https://postcutoff.com/e/2026-08-19-caratheodory-loewner-smooth-counterexamples-alpoge-claude/)