The band-gaps of CsPbI$_3$ perovskite nanocrystals are measured by absorption spectroscopy at cryogenic temperatures. Anomalous band-gap shifts are observed in CsPbI$_3$ nanocubes and nanoplatelets, which are modeled accurately by band-gap renormalization due to lattice vibrational modes. We find that decreasing dimensionality of the CsPbI$_3$ lattice in nanoplatelets greatly reduces electron-phonon coupling, and dominant out-of-plane quantum confinement results in a homogeneously broadened absorption lineshape down to cryogenic temperatures. An absorption tail forms at low-temperatures in CsPbI$_3$ nanocubes, which we attribute to shallow defect states positioned near the valence band-edge.
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