Positivity principle for equivalent norms (Q2736047)

For a separable Banach space \(X\), the norm is called a p.p. (positivity principle) norm, if for any 2 finite Borel measures \(\mu\), \(\nu\) on \(X\) with \(\mu(B)\leq \nu(B)\) for all balls of radius 1 one has \(\mu\leq\nu\). The main result states that in the set of all norms on \(X\) equivalent with the given one the p.p.-norms are dense with respect to the Hausdorff metric \(d(B_1, B_2)\), \(B_j= \{\|x\|_j\leq 1\}\).NEWLINENEWLINENEWLINEThis is done via \(X\) with a basis and convex open sets. If \(X\) has a basis, then the p.p.-norms are even of second category in this set of all equivalent norms, with respect to a modified Hausdorff metric.NEWLINENEWLINENEWLINEFinally, the locally uniformly convex Day norm on \(c_0\) is shown to satisfy \(\mu\leq\nu\) whenever \(\mu(B)\leq \nu(B)\) for all balls with radius \(<1\), \(\mu\) and \(\nu\) finite Borel measures on \(c_0\); on \(\ell^2\) equivalent uniformly smooth p.p.-norms are constructed which are dense with respect to the Hausdorff metric (the \(\ell^2\)-norm is not even p.p. with respect to all balls of radius \(\leq 1\)).