Convex Optimization and Parsimony of $L_p$-balls Representation

In the family of unit balls with constant volume we look at the ones whose algebraic representation has some extremal property. We consider the family of nonnegative homogeneous polynomials of even degree $p$ whose sublevel set $\mathbf{G}=\{\mathbf{x}: g(\mathbf{x})\leq 1\}$ (a unit ball) has the same fixed volume and want to find in this family the polynomial that minimizes either the parsimony-inducing $\ell_1$-norm or the $\ell_2$-norm of its vector of coefficients. Equivalently, among all degree-$p$ polynomials of constant $\ell_1$- or $\ell_2$-norm, which one minimizes the volume of its level set $\mathbf{G}$? We first show that in both cases this is a convex optimization problem with a unique optimal solution $g^*_1$ or $g^*_2$, respectively. We also show that $g^*_1$ is the $L_p$-norm polynomial $\mathbf{x}\mapsto\sum_{i=1}^n x_i^{p}$, thus recovering a parsimony property of the $L_p$-norm polynomial via $\ell_1$-norm minimization. This once again illustrates the power and versatility of the $\ell_1$-norm relaxation strategy in optimization when one searches for an optimal solution with parsimony properties. Next we show that $g^*_2$ is not sparse at all (and thus differs from $g^*_1$) but is still a sum of $p$-powers of linear forms. In fact, and surprisingly, for $p=2,4,6,8$, we show that $g^*_2=(\sum_ix_i^2)^{p/2}$, whose level set is the Euclidean (i.e., the $L_2$-norm) ball. We also characterize the unique optimal solution of the same problem where one searches for a sum of squares homogeneous polynomial that minimizes the (parsimony-inducing) nuclear norm of its associated (positive semidefinite) Gram matrix, hence aiming at finding a solution which is a sum of a few squares only. Again for $p=2,4$ the optimal solution is $(\sum_ix_i^2)^{p/2}$, whose level set is the Euclidean ball, and when $p\in\,4\mathbb{N}$, this is also true when $n$ is sufficiently large. Finally, we also extend these results to generalized homogeneous polynomials, which include $L_p$-norms when $0<p$ is rational.