Existence and a partial characterization of perfect splines of minimum norm, related to totally positive kernels, are obtained by a unified method applicable to the class of monotone norms (norms for which |f(x)|≤|g(x)| implies ∥f∥≤∥g||). The method is based on the duality between this problem and best L1-approximation, which provides a pointwise improvement theorem for perfect splines. Similar results are obtained for norms induced by inner products by the equivalence between this case and the self-dual case of perfect splines of minimum L1-norm. The knots and zeros of the minimal perfect splines are then used in choosing best tensor-product approximations to totally positive kernels in a norm which is a tensor product of a monotone norm and the L1-norm. Also n-widths and optimal spaces in the sense of Kolmogorov and Gelfand are obtained for integral operators with totally positive kernels via the minimal perfect splines. These results generalize known results for the Lp-norms, 1 ≤ p ≤ ∞, to monotone norms and to norms induced by inner products.