Standard analysis of compositional data under the assumption that the Aitchison geometry holds assumes a uniform distribution as reference measure of the space. Weighting of parts can be done changing the reference measure. The changes that appear in the algebraic-geometric structure of the simplex are analysed, as a step towards understanding the implications for elementary statistics of random compositions. Some of the standard tools in exploratory analysis of compositional data analysis, such as center, variation matrix and biplots are studied in some detail, although further research is still needed. The main conclusion is that down-weighting some parts is approaching the geometry of the corresponding subcomposition, thus preserving a kind of coherence between standard and down-weighted analyses