  Variable selection is considered in the setting of supervised binary
classification with functional data $\{X(t),\ t\in[0,1]\}$. By "variable
selection" we mean any dimension-reduction method which leads to replace the
whole trajectory
  $\{X(t),\ t\in[0,1]\}$, with a low-dimensional vector
$(X(t_1),\ldots,X(t_k))$ still keeping a similar classification error. Our
proposal for variable selection is based on the idea of selecting the local
maxima $(t_1,\ldots,t_k)$ of the function ${\mathcal V}_X^2(t)={\mathcal
V}^2(X(t),Y)$, where ${\mathcal V}$ denotes the "distance covariance"
association measure for random variables due to Sz\'ekely, Rizzo and Bakirov
(2007). This method provides a simple natural way to deal with the relevance
vs. redundancy trade-off which typically appears in variable selection. This
paper includes
  (a) Some theoretical motivation: a result of consistent estimation on the
maxima of ${\mathcal V}_X^2$ is shown. We also show different theoretical
models for the underlying process $X(t)$ under which the relevant information
in concentrated in the maxima of ${\mathcal V}_X^2$.
  (b) An extensive empirical study, including about 400 simulated models and
real data examples, aimed at comparing our variable selection method with other
standard proposals for dimension reduction.
