Yesterday I also circulated this question to my colleagues and one more example was brought to my attention.
https://arxiv.org/abs/1905.11968
https://arxiv.org/abs/1905.11877
(both of them appeared on arXiv on 28 May 2019)
They solve the following problem: one receives a convex body $K_{0}$ in $\mathbb{R}^{n}$ and chooses a point $x_{0} \in K_{0}$, then one receives another convex body $K_{1}$, and choose a point $x_{1} \in K_{1}$ etc. The goal is after $N$ steps to make the total distance between these points $\sum_{i=1}^{N}\| x_{j}-x_{j-1}\|$ minimal. They both find an algorithm such that $\frac{\sum_{i=1}^{N}\| x_{j}-x_{j-1}\|}{\sum_{i=1}^{N}\| \tilde{x}_{j}-\tilde{x}_{j-1}\|} \leq O(\sqrt{n \ln T})$, where $\sum_{i=1}^{N}\| \tilde{x}_{j}-\tilde{x}_{j-1}\|$ is the smallest distance one could get if one would in advance knew positions and shapes of all convex bodies $\{K_{j}\}_{j=0}^{N}$.