classifier ${\widehat}{\Phi}$ must satisfy (omitting again logarithmic factors and constant factors depending only on $s$ and $R$): $$\sup_{\substack{f,g \in {\mathcal{H}_s}(R) \\ \|f-g\| {\geqslant}\Delta}} \left\{ \mathcal{R}_{f,g}({\widehat}\Phi) - \inf_{\Phi} \mathcal{R}_{f,g}(\Phi)  \right\} \gtrsim \left\{\begin{array}{ll}
        n^{-\frac{s}{2s+1}} & \textrm{if } \Delta \lesssim n^{-\frac{s}{2s+1}} \\[0.2em]
        \displaystyle \frac{1}{\Delta} n^{-\frac{2s}{2s+1}} & \textrm{if } \Delta \gtrsim n^{-\frac{s}{2s+1}}
    \end{array}\right.$$

-   Finally, in Section \[s:knn\], we show that the well-known $k$-nearest neighbors rule tuned in a classical and optimal way (see, [e.g.]{}, [@S12; @GKM16]) is far from optimality in our specific functional setting.

    For any threshold (dimension) ${\widehat}{d} \in \mathbb{N}^*$ based on a sample-splitting policy, and for the optimal choice $k_n^{opt}({\widehat}{d})=\lfloor n^{4/(4+{\widehat}{d})}\rfloor$, the ${\widehat}{d}$-dimensional $k_n^{opt}({\widehat}{d})$-nearest neighbors classifier $\Phi_{\textrm{NN}}$ suffers a logarithmic excess risk in the worst case: $$\sup_{f,g \in {\mathcal{H}_s}(r)} \Bigl\{ \mathcal{R}_{f,g}(\Phi_{\textrm{NN}}) - \inf_{\Phi} \mathcal{R}_{f,g}(\Phi) \Bigr\} \gtrsim \log(n)^{-2s} \,.$$

 \
Most proofs are postponed to Appendix \[s:A\] (for the upper bounds) and to Appendix  \[s:minimaxlowerbound\] (for the lower bounds).\
[**Other useful notation.**  ]{} We denote the joint distribution of the pair $((X(t))_{t\in[0,1]},Y)$ by $\mathbb{P}_{X,Y}$, and write $\mathbb{P}_{\otimes^n} = P_{X,Y}^{\otimes n}$ for the joint distribution of the sample $(X_i,Y_i)_{1 {\leqslant}i {\leqslant}n}$. For notational convenience, the measure $\mathbb{P}$ will alternatively stand for $\mathbb{P} = \mathbb{P}_{X,Y} \times \mathbb{P}_{\otimes^n}$ (we integrate over both the sample 