# A convex function satisfying the Lojasiewicz inequality but failing the gradient conjecture both at zero and infinity

arXiv: https://arxiv.org/abs/2102.05342 · HTML: https://mlsanity.com/paper/2102.05342/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Feb 2021.

## Built on by (most cited first, 3 of 3)

- [SAD Neural Networks: Divergent Gradient Flows and Asymptotic Optimality via o-minimal Structures](https://mlsanity.com/paper/2505.09572/) (2025)
- [A qualitative difference between gradient flows of convex functions in finite- and infinite-dimensional Hilbert spaces](https://mlsanity.com/paper/2310.17610/) (2023)
- [Convergence Rates of Stochastic Zeroth-order Gradient Descent for L ojasiewicz Functions](https://mlsanity.com/paper/2210.16997/) (2022)

## Abstract (arXiv)

We construct an example of a smooth convex function on the plane with a strict minimum at zero, which is real analytic except at zero, for which Thom's gradient conjecture fails both at zero and infinity. More precisely, the gradient orbits of the function spiral around zero and at infinity. Besides, the function satisfies the Lojasiewicz gradient inequality at zero.