4 problems

Let $N>0$, let $I$ be the interval under consideration, and let $f:[0,N]\to[0,N]$ be a strictly convex function such that $f\in C^3([0,N])$ and $f^{(3)}(x)\neq 0$ for all…

Let $\gamma:\mathbb{R}\to\mathbb{R}_{m-1}[u]$ be a convex curve, meaning that any $m$ distinct points along $\gamma$ are linearly independent. Let $z_1,\dots,z_{d(m-d)}$ be distinc…

Let $\mathcal{L}\mathbb S^3(\mathbf{1},-\mathbf{1})$ be the space of locally convex curves on the 3-sphere with the indicated endpoint data, let $L$ denote the associated map to cu…

Two-tripod conjecture. Every smooth closed convex curve in the spherical or hyperbolic geometry has at least two tripod configurations.