  The spectral measure plays a key role in the statistical modeling of
multivariate extremes. Estimation of the spectral measure is a complex issue,
given the need to obey a certain moment condition. We propose a Euclidean
likelihood-based estimator for the spectral measure which is simple and
explicitly defined, with its expression being free of Lagrange multipliers. Our
estimator is shown to have the same limit distribution as the maximum empirical
likelihood estimator of J. H. J. Einmahl and J. Segers, Annals of Statistics
37(5B), 2953--2989 (2009). Numerical experiments suggest an overall good
performance and identical behavior to the maximum empirical likelihood
estimator. We illustrate the method in an extreme temperature data analysis.
