  Linear bandit algorithms yield $\tilde{\mathcal{O}}(n\sqrt{T})$ pseudo-regret
bounds on compact convex action sets $\mathcal{K}\subset\mathbb{R}^n$ and two
types of structural assumptions lead to better pseudo-regret bounds. When
$\mathcal{K}$ is the simplex or an $\ell_p$ ball with $p\in]1,2]$, there exist
bandits algorithms with $\tilde{\mathcal{O}}(\sqrt{nT})$ pseudo-regret bounds.
Here, we derive bandit algorithms for some strongly convex sets beyond $\ell_p$
balls that enjoy pseudo-regret bounds of $\tilde{\mathcal{O}}(\sqrt{nT})$,
which answers an open question from [BCB12, \S 5.5.]. Interestingly, when the
action set is uniformly convex but not necessarily strongly convex, we obtain
pseudo-regret bounds with a dimension dependency smaller than
$\mathcal{O}(\sqrt{n})$. However, this comes at the expense of asymptotic rates
in $T$ varying between $\tilde{\mathcal{O}}(\sqrt{T})$ and
$\tilde{\mathcal{O}}(T)$.
