On a Generalized Best Approximation Problem
On a Generalized Best Approximation Problem
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DOI：
10.1006/jath.1998.3177
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发表时间：
1998-07
影响因子：
0.9
通讯作者：
F. Blasi;J. Myjak
中科院分区：
文献类型：
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作者：
F. Blasi;J. Myjak
设C是Banach空间E的闭有界凸子集，其原点E为内点，令pC表示关于C的明可夫斯基泛函。给定一个闭集X?E和一个点u?E，我们考虑一个最小化问题minC(u, X)，它包括证明点x?X的存在性，使得pC(x?u)=?C(u, X)，其中?C(u, X)=inf{pC(x?u)?x?X}。如果这样的点是唯一的，并且满足条件 limn?+∞pC(xn?u)=?C(u, X) 的每个序列 {xn}?X 都收敛到该点，则最小化问题 min(u, X) 称为适定问题。在关于topC的凸模严格为正的假设下，我们证明对于每个闭子集XofE，最小化问题minC(u, X)适定的allu?E的setEo(X)是E的残差子集。事实上我们展示了更多，即集合E\Eo(X)在E中是？-多孔的。此外，我们证明对于大多数闭有界子集XofE，集合E\Eo(X)在E中是稠密的。
LetCbe a closed bounded convex subset of a Banach spaceEwhich has the origin ofEas an interior point and letpCdenote the Minkowski functional with respect toC. Given a closed setX?Eand a pointu?Ewe consider a minimization problem minC(u, X) which consists in proving the existence of a pointx?Xsuch thatpC(x?u)=?C(u, X), where?C(u, X)=inf{pC(x?u)?x?X}. If such a point is unique and every sequence {xn}?Xsatisfying the condition limn?+∞pC(xn?u)=?C(u, X) converges to this point, the minimization problem min(u, X) is called well posed. Under the assumption that the modulus of convexity with respect topCis strictly positive, we prove that for every closed subsetXofE, the setEo(X) of allu?Efor which the minimization problem minC(u, X) is well posed is a residual subset ofE. In fact we show more, namely that the setE\Eo(X) is?-porous inE. Moreover, we prove that for most closed bounded subsetsXofE, the setE\Eo(X) is dense inE.