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In this paper, we generalized the variational definition of median numbers to median shapes. We represent shapes with currents and use flat norms as the distances between currents. Under this setting, the median shapes problem becomes an optimization problem and the median shapes will be the shapes that minimize the sum of distances from itself to all other input currents. In order to obtain some regularity result, we also define the mass regularized median which adds an extra mass regularization term in the optimization problems. In codimension 1 case, we proved the existence for both median and mass regularized median given a specific class of input currents. And then we showed the support of the differences between input currents and medians lie inside the envelope and for mass regularized medians the support of the differences are contained in the $\epsilon$ extension of the envelope. For codimension 2, we presented an interesting case for curves in $\R^3$ with shared boundaries and their tangent spaces satisfy the ``book condition". Some computational results and codes are also provided online.