On the distribution of the norm for a Gaussian measure (Q796885)

Let E be an infinite dimensional Banach space with norm \(\| \cdot \|\), and \(\mu\) a centered Gaussian measure on E. It is known that the distribution of \(\| \cdot \|\) for \(\mu\) has a bounded continuous density with respect to Lebesgue measure on each interval \([t,\infty [\) for \(t>0\). Whether the density is bounded on all \({\mathbb{R}}^+\) closely depends on the shape of the unit ball. We show here that for each \(\epsilon>0\), there is a norm N that is \((1+\epsilon)\)-equivalent to \(\| \cdot \|\) and a centered Gaussian measure \(\mu\) on E such that the distribution of N(.) has unbounded density with respect to Lebesgue measure. In other words, there exists no constant C such that \(\mu\) ([x; \(\alpha \leq N(x)\leq \alpha +\delta])\leq C\delta\) for each \(\alpha,\delta>0\).