8 problems

Let $D$ be a bounded convex domain in a finite-dimensional real vector space, endowed with its Hilbert metric $d_H$, and let $f:D\rightarrow D$ be a fixed-point-free nonexpansive m…

Reflexive-space fixed point conjecture. Every reflexive Banach space has the fixed point property for nonexpansive maps. Equivalently, if $X$ is reflexive and $C\subset X$ is close…

Let $U$ and $V$ be subspaces and let $P_U$ and $P_V$ denote the corresponding orthogonal projections. Consider the composition … where $R_U$ and $R_V$ are the associated reflected…

Let $X$ be a one-dimensional or higher-dimensional setting with a vector $e$ and an operator $A_e$ whose degree of cyclic monotonicity is being considered. The endpoint cases are t…

Convergence conjecture. Even though the cited dimension theorem is not applicable, the sequence

Let $E$ be a smooth and uniformly convex Banach space of dimension two. Let $K\subseteq E$ be nonempty, closed, bounded, and convex, and let $f\colon K\to E$ be nonexpansive with…

Convergence conjecture. The sequence $(Q_n(x))_{n\in\mathbb{N}}$ actually converges.

Karlsson–Nussbaum conjecture. There exists a convex set $\Lambda$ in $\partial\Omega$ such that