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10 pages, 6 figures, 2 tables. ; In this work we analyze and compare the free volume of a polymer system poly(vinyl methylether) (PVME) at 300 K obtained by the two direct but different approaches: Positron annihilation lifetime spectroscopy (PALS) and computer simulations. The free volume is calculated from the simulated cells of PVME by means of numerical methods based on grid scanning and probing the structure with a probe of a given radius RP. The free-volume structure was found to be percolated for small probes at RP=0.53 Å. As the probe radius increases, the cavity structure breaks into isolated cavities, reaching a maximum of the cavity number at RP=0.78 Å. We further develop methods for a geometrical analysis of the free-volume cavities by considering their shape. The geometrical computations show that the cavities have elongated shape with side-to-length ratio corresponding to approximately 1:0.55 and with an average length of 6 Å. Based on the overlap between the computed cavities and simplified geometrical representations, the best match of the cavity shape is obtained for the approximation to the ellipsoidal shape (overlap on 84.4%). A match with other examined shapes follows the sequence: ellipsoid>cylinder>bar>sphere>cube. Finally, the computed geometrical parameters are used as input parameters into the quantum-mechanical models for the orthopositronium (o-Ps) lifetime in various free-volume hole geometries. Comparison with the experimental data gives support for two ideas about the existence of an o-Ps particle in the polymeric matrix: (i) the positronium cannot localize in a portion of very small cavities; (ii) and in the case of the percolated cavities, several o-Ps particles occupy some subcavities in the same cavity. Additionally, radial distribution functions of the free volume indicate the existence of two kinds of free volume, a structured one, corresponding to interstitial spaces along the polymer chain, and the so-called "bulk free volume," distributed randomly in the structure. PALS measurements seem to be mainly related with this bulk free volume. The cavities represented by the idealized geometries are visualized in three-dimensional space providing a unique representation on the free-volume structures. ; This work was supported by the VEGA Agency, Slovakia under Grant No. 2/0014/09(J.B.)and Project No. MAT2007-63681 (Spanish Ministry of Education), Grant No. IT-436-07 (Basque Government) and Project No. 2007SK0008 between Slovak Academy of Sciences (SAS) and Spanish Research Council (CSIC) Support from the project Spanish MEC Grant No. CSD2006-53 is also acknowledged. ; Peer reviewed