Finite sets of piecewise linear inequalities do not characterize zonoids (Q1383583)

A piecewise linear inequality (PLI) is a relation of the form \[ \sum^k_{i=1} c_i\left | \sum^n_{j=1} a_{ij} w_j \right| \geq 0 \] which holds for all real variables \(w_1, \dots, w_n\), with fixed real \(c_i\) and \(a_{ij}\). It is known that a centered convex body \(Z\subset \mathbb{R}^d\) \((d\geq 3)\) is a zonoid if and only if each PLI holds in \(\mathbb{R}^d\), with the absolute value replaced by the norm given by the support function of \(Z\), and with \(w_1, \dots, w_n\) being arbitrary elements of \(\mathbb{R}^d\). It has repeatedly been asked in the literature whether a single PLI is sufficient to characterize the zonoids among all centered convex bodies. The present paper gives a strong negative answer: no finite set of PLIs is sufficient to characterize the zonoids in \(\mathbb{R}^d\). In particular, even in \(\mathbb{R}^3\) the Hlawka inequality does not characterize zonoids. This answers a question posed by P. Goodey and W. Weil.