论文标题
从子空间的连续Quasiconvex函数的可扩展性
Extendability of continuous quasiconvex functions from subspaces
论文作者
论文摘要
令$ y $为拓扑矢量空间$ x $的子空间，以及$ a \ subset x $与$ y $相交的开放式凸套装。我们说，如果在$ a \ cap y $上的每个连续的Quasiconvex [连续凸面]函数中，属性$（QE）$ [property $（ce）$]承认连续定义了$ a $ $ a $的Quasiconvex [连续凸面]扩展。我们研究$（QE）$和$（CE）$ properties之间的关系，证明$（QE）$总是暗示$（CE）$，并且在适当的假设下（例如，如果$ x $是标准的空间，并且$ y $是$ x $的封闭子空间），则两个物业相当于。 通过将$（QE）$和$（CE）$属性之间的先前含义与有关属性$（CE）$的已知结果之间的含义结合在一起，我们就可以获得有关Quasiconvex连续功能的扩展的一些新的积极结果。特别是，我们将\ cite {deqex}中包含的结果概括为无限差分可分离情况。此外，我们还立即获得了$（QE）$不存在的示例的存在。
Let $Y$ be a subspace of a topological vector space $X$, and $A\subset X$ an open convex set that intersects $Y$. We say that the property $(QE)$ [property $(CE)$] holds if every continuous quasiconvex [continuous convex] function on $A\cap Y$ admits a continuous quasiconvex [continuous convex] extension defined on $A$. We study relations between $(QE)$ and $(CE)$ properties, proving that $(QE)$ always implies $(CE)$ and that, under suitable hypotheses (satisfied for example if $X$ is a normed space and $Y$ is a closed subspace of $X$), the two properties are equivalent. By combining the previous implications between $(QE)$ and $(CE)$ properties with known results about the property $(CE)$, we obtain some new positive results about the extension of quasiconvex continuous functions. In particular, we generalize the results contained in \cite{DEQEX} to the infinite-dimensional separable case. Moreover, we also immediately obtain existence of examples in which $(QE)$ does not hold.
