Having a convex cone K in an infinite-dimensional real linear space X , Adán and Novo stated (in J Optim Theory Appl 121:515–540, 2004) that the relative algebraic interior of K is nonempty if and only if the relative algebraic interior of the positive dual cone of K is nonempty. In this paper, we show that the direct implication is not true even if K is closed with respect to the finest locally convex topology $$\tau _{c}$$ τ c on X , while the reverse implication is not true if K is not $$\tau _{c}$$ τ c -closed. However, in the main result of this paper, we prove that the latter implication is true if the algebraic interior of the positive dual cone of K is nonempty; the general case remains an open problem. As a by-product, a result about separation of cones is obtained that improves Theorem 2.2 of the work mentioned above.

中文翻译：

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关于实线性空间中相对实心凸锥

在无限维实线性空间 X 中具有凸锥 K，Adán 和 Novo 指出（在 J Optim Theory Appl 121:515–540, 2004 中）K 的相对代数内部是非空的当且仅当相对代数内部K 的正对偶锥的 非空。在本文中，我们表明，即使 K 相对于 X 上最好的局部凸拓扑 $$\tau _{c}$$ τ c 是封闭的，直接蕴涵也不是真的，而如果 K 则相反的蕴涵不是真的不是 $$\tau _{c}$$ τ c -关闭。然而，在本文的主要结果中，我们证明了如果 K 的正对偶锥的代数内部非空，则后一个蕴涵成立；一般情况仍然是一个悬而未决的问题。作为副产品，获得了关于锥体分离的结果，该结果改进了上述工作的定理 2.2。