Low-Degree Polynomial Approximation of the Cross-Polytope

Abstract

We determine the degree-distortion tradeoff for polynomial approximation of the $d$-dimensional cross-polytope $B_1^d$. For every $1\le t\le d$, every globally nonnegative degree-$2t$ form that is positive away from the origin has multiplicative sandwich distortion at least $(2e)^{-1/2}\sqrt{d/t}$, while an explicit sum-of-squares (SoS) form achieves distortion at most $(2e)^{1/2}\sqrt{d/t}$. Hence both the nonnegative-form and SoS optima are $\Theta(\sqrt{d/t})$, and degree $\Theta(d)$ is necessary and sufficient for constant distortion. The lower bound is representation-free. Averaging over signed permutations and evaluating on flat points of the $\ell_1$ sphere reduces every candidate to a support-size profile $V(k)=k^{-2t}Q(k)$ with $\deg Q\le t$ and $Q(0)=0$. After the substitution $u=1/k$, Lagrange interpolation shows that this degree budget cannot keep the profile nearly constant across $d$ support scales. A matching SoS construction averages even powers of sign-vector facet normals and reduces the upper bound to a Rademacher moment. We also isolate a weighted reciprocal-grid lemma, derive consequences for $\ell_p$ balls, and contrast the polar cube. For a general symmetric polytope, weighted facet powers yield a one-sided certificate whose boundary-floor objective is concave and whose worst-direction oracle reduces to convex dual-norm problems; at degree two, its optimizers recover classical optimal design and the John ellipsoid. This is an oracle-model certificate optimization, not an end-to-end complexity result or a characterization of the full SoS optimum.