  We present a new summary statistic for weak lensing observables, higher than
second order, suitable for extracting non-Gaussian cosmological information and
inferring cosmological parameters. We name this statistic the 'starlet
$\ell_1$-norm' as it is computed via the sum of the absolute values of the
starlet (wavelet) decomposition coefficients of a weak lensing map. In
comparison to the state-of-the-art higher-order statistics -- weak lensing peak
counts and minimum counts, or the combination of the two -- the $\ell_1$-norm
provides a fast multi-scale calculation of the full void and peak distribution,
avoiding the problem of defining what a peak is and what a void is: The
$\ell_1$-norm carries the information encoded in all pixels of the map, not
just the ones in local maxima and minima. We show its potential by applying it
to the weak lensing convergence maps provided by the MassiveNus simulations to
get constraints on the sum of neutrino masses, the matter density parameter,
and the amplitude of the primordial power spectrum. We find that, in an ideal
setting without further systematics, the starlet $\ell_1$-norm remarkably
outperforms commonly used summary statistics, such as the power spectrum or the
combination of peak and void counts, in terms of constraining power,
representing a promising new unified framework to simultaneously account for
the information encoded in peak counts and voids. We find that the starlet
$\ell_1$-norm outperforms the power spectrum by $72\%$ on M$_{\nu}$, $60\%$ on
$\Omega_{\rm m}$, and $75\%$ on $A_{\rm s}$ for the Euclid-like setting
considered; it also improves upon the state-of-the-art combination of peaks and
voids for a single smoothing scale by $24\%$ on M$_{\nu}$, $50\%$ on
$\Omega_{\rm m}$, and $24\%$ on $A_{\rm s}$.
