High pressure x-ray diffraction (XRD) measurements on RMnO3 (R=Dy, Ho and Lu) reveals that varying structural changes occurs for different R ions. Large lattice changes (orthorhombic strain) occur in DyMnO3 and HoMnO3 while the Jahn-Teller (JT) distortion remains stable. On the other hand, in LuMnO3, Mn-O bond distortions are observed in the region 4-8 GPa with the broad minimum in the JT distortion. High pressure IR measurements indicate that a phonon near 390 cm-1 corresponding to the complex motion of the Mn and O ions changes anomalously for LuMnO3. It softens in the 4-8 GPa region, which is consistent with the structural change in Mn-O bonds and then hardens at high pressures. By contrast, the phonons continuously harden with increasing pressure for DyMnO3 and HoMnO3. DFT calculations show that the E-phase LuMnO3 is the most stable phase up to the 10 GPa pressure examined. Simulations indicate that the distinct structural change under pressure in LuMnO3 can possibly be used to optimize the electric polarization by pressure/strain.