Two dimensional zonoids and Chebyshev measures
An \(\mathbb{R}^2\)-valued measure \(\mu= (\mu_1,\mu_2)\) on \([0,1]\) (with a \(\sigma\)-field) is a Chebyshev measure if it is non-atomic and each \(|\mu|^{\otimes 2}\)-non-negligible measurable subset of \([0,1]\times[0, 1]\) has positive \(\text{det }\mu\) measure, where \(\text{det }\mu= \mu_1\otimes\mu_2- \mu_2\otimes\mu_1\). The authors prove that a set \(K\subset\mathbb{R}^2\) is the range of some Chebyshev measure if and only if \(K\) is strictly convex, compact, centrally symmetric, and contains \((0,0)\) in its boundary. Further results for \(\mathbb{R}^2\)-valued measures \(\mu\) are: a condition under which \(\mu\) is the difference of two Chebyshev measures, a condition on the density of \(\mu\) (with respect to its variation measure \(|\mu|\)) that is necessary for the strict convexity of the range of \(\mu\), and a characterization of the Chebyshev measures.
scientific article; zbMATH DE number 1053436