The "Tip" of the Red Giant Branch as a distance indicator: theoretical calibration and the value of H_0

Updated theoretical relations for the run of the bolometric and I magnitude of the Tip of the Red Giant Branch (TRGB) with respect to the metallicity are provided. An analogous relation for the V magnitude of the Zero Age Horizontal Branch at the RR Lyrae instability strip is also provided. A comparison has been performed among our ZAHB and TRGB distances, the Cepheid distance scale by Madore & Freedman (1991) and the HIPPARCOS distances set by local subdwarfs with accurate parallax determinations. The application of our TRGB distance scale to NGC3379 provides a distance to the Leo I group that is about 8% higher than the one obtained by Sakai et al. (1997) adopting the TRGB brightness calibration by Da Costa & Armandroff (1990). Our distance to the Leo I group, coupled with the relative distance Coma cluster-Leo I, provides a determination of H_0 at the Coma cluster: H_0=64_{-9}^{+10} Km s^{-1} Mpc^{-1}.