Moment inequalities for functionals of the Brownian convex hull (Q1196925)

Let \({\mathcal E}^ d\) be the collection of all convex subsets of the \(d\)- dimensional Euclidean space \(R^ d\). Let \(\varphi\geq 0\) be a function on \({\mathcal E}^ d\) with the properties (1) \(\varphi(A)\leq\varphi(B)\) for \(A\subseteq B\) from \({\mathcal E}^ d\), (2) \(\varphi(rc)=r^ \alpha\varphi(c)\) for \(c\in {\mathcal E}^ d\), \(r>0\) and some \(\alpha>0\). Let \(\mathbb{C}(t)\), \(t\geq 0\), be the convex hull of the set points \(X([0,t])=\{x\in R^ d\mid\exists s<t\) such that \(X(s)=x\}\), where \(X(t)\), \(t\geq 0\), is a \(d\)-dimensional Brownian process. The main result of the paper is: For all stopping times \(\tau\) with respect to the natural filtration of \(X(t)\), \(t\geq 0\), there exist constants \(c_ 1>0\) and \(c_ 2>0\) such that \[ c_ 1 E \tau^{\alpha/2}\leq E \varphi(\mathbb{C}(\tau))\leq c_ 2 E \tau^{\alpha/2}. \] The main idea used in proving this result is due to \textit{D. L. Burkholder} [Ann. Probab. 1, 19-42 (1973; Zbl 0301.60035)]. He proved the following inequality \[ c_ 1 E \tau^{r/2}\leq E\{\max_{s\leq\tau} X(t)\}^ r\leq c_ 2 E \tau^{r/2},\quad r>0, \] for a standard linear Brownian motion \(X(t)\), \(t\geq 0\).