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Whereas the norm of a vector measures amplitude (and is a 1-homogeneousfunction), sparsity is measured by the 0-homogeneous l0 pseudonorm, whichcounts the number of nonzero components. We propose a family of conjugaciessuitable for the analysis of 0-homogeneous functions. These conjugacies arederived from couplings between vectors, given by their scalar product dividedby a 1-homogeneous normalizing factor. With this, we characterize the bestconvex lower approximation of a 0-homogeneous function on the unit ''ball'' ofa normalization function (i.e. a norm without the requirement ofsubadditivity). We do the same with the best convex and 1-homogeneous lowerapproximation. In particular, we provide expressions for the tightest convexlower approximation of the l0 pseudonorm on any unit ball, and we show that thetightest norm which minorizes the l0 pseudonorm on the unit ball of any lp-normis the l1-norm. We also provide the tightest convex lower convex approximationof the l0 pseudonorm on the unit ball of any norm.

Best Convex Lower Approximations of the l 0 Pseudonorm on Unit Balls