We consider the estimation of the number N of present species in a given area at a given time, based on the abundances of species that have been observed. We adopt a nonparametric approach where the true abundance distribution p is only supposed to be convex. A definition for convex abundance distributions is proposed. A least-squares estimate of the truncated version of p under the convexity constraint is used. Two estimators of N are deduced, the asymptotic distribution of which are derived. We propose three different procedures, including a bootstrap one, to obtain a confidence interval for N and a standard error for its estimator. The performances of the estimators are assessed in a simulation study and compared with competitors. The proposed method is illustrated on several examples.