In this paper, we define almost convex space. Let $$T:X\rightarrow Y$$ be a linear bounded operator. This paper shows that: (1) If X is almost convex and 2-strictly convex, Y is a Banach space, D(T) is closed, N(T) is an approximatively compact Chebyshev subspace of D(T) and R(T) is a 2-Chebyshev hyperplane of Y, then there exists a homogeneous selection $${T^\sigma }$$ of $${T^\partial }$$ such that continuous points of $$T^\sigma $$ is dense on Y. (2) If X is locally uniformly convex, Y is reflexive, D(T) is closed, N(T) is a proximinal subspace of D(T) and R(T) is a closed hyperplane of Y, then $$T^{\partial }$$ is single-valued, homogeneous and continuous on Y. The results are a perfect answer to the open problem posed by Nashed and Votruba (Bull Am Math Soc 80:831–835, 1974).

中文翻译：

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Banach空间中集值度量广义逆的近似凸性和连续选择

在本文中，我们定义了几乎凸空间。令 $$T:X\rightarrow Y$$ 是一个线性有界算子。本文表明： (1) 如果X 几乎是凸的并且是2-严格凸的，则Y 是Banach 空间，D(T) 是封闭的，N(T) 是D(T) 和R( T) 是 Y 的 2-Chebyshev 超平面，那么存在 $${T^\sigma $$$${T^\partial }$$ 的齐次选择 $${T^\sigma }$$ 使得 $$T^\sigma $ 的连续点$ 在 Y 上是稠密的。 (2) 如果 X 是局部一致凸的，Y 是自反的，D(T) 是封闭的，N(T) 是 D(T) 的近邻子空间，R(T) 是Y，则 $$T^{\partial }$$ 是 Y 上的单值、齐次和连续的。结果是对 Nashed 和 Votruba 提出的开放问题的完美答案（Bull Am Math Soc 80：831–835， 1974）。