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With inspiration from the K\"ahler geometry, we introduce a metric structureon the energy class, $\mathcal{E}_{1,m}$, of $m$-subharmonic functions withbounded energy and show that it is complete. After studying how the metricconvergence relates to the accepted convergences in thisCaffarelli-Nirenberg-Spruck model, we end by constructing geodesics in asubspace of our complete metric space.

Geodesics in the space of $m$-subharmonic functions with bounded energy