Maps of Convex Sets in Hilbert Spaces
convex sets | finite difference schemes | convex-valued maps
Let $\H$ be a separable Hilbert space, $\U\subset \H$ an open convex subset, and $f:\U\to \H$ a smooth map. Let $\Om$ be an open convex set in $\H$ with $\bOm\subset \U$, where $\bOm$ denotes the closure of $\Om$ in $\H$. We consider the following questions. First, in case $f$ is Lipschitz, find sufficient conditions such that for $\ve>0$ sufficiently small, depending only on $\Lip(f)$, the image of $\Om$ by $I\pm\ve f$, $(I\pm\ve f)(\Om)$, is convex. Second, suppose $df(u):\H\to\H$ is symmetrizable with $\s(df(u))\subset(0,\infty)$, for all $u\in\U$, where $\s(df(u))$ denotes the spectrum of $df(u)$. Find sufficient conditions so that the image $f(\Om)$ is convex. We establish results addressing both questions illustrating our assumptions and results with simple examples. We apply them to finite difference methods for approximating solutions of nonlinear ordinary differential equations in Hilbert spaces and also discuss the invariance of convex-valued maps from measure spaces into Hilbert spaces under certain nonlinear integral operators.