1
$\begingroup$

Let $C$ be a closed convex cone in a Hausdorff locally convex topological vector space $E$ and let $y \in E$. I wonder a condition under which the conical hull of $C \cup \left\{ y \right\}$, i.e., the set $\left\{ x+\lambda y: x \in C, \lambda \geq 0 \right\}$, is closed.

Any suggestions will be very much appreciated.

$\endgroup$
1
$\begingroup$

Let $C^\ominus = \{ u\in E^* \,|\, \langle u,C\rangle \leq 0\}$ and let $H = \{u\in E^* \,|\, \langle u,-y\rangle \leq 0\}$.

If the interior of $C^\ominus$ meets $H$, or if $C^\ominus$ meets the interior of $H$, then we have the sum rule $\partial (\iota_{C^\ominus}+\iota_{H})= \partial \iota_{C^\ominus} + \partial \iota_{H}$. In that case, evaluating at $0$, gives

$$ (C^\ominus\cap H)^\ominus = C+H^\ominus = C+\mathbb{R}_+y\;\;\text{is closed.}$$

Another case arises when $C$ is a polyhedral cone. All the above is true in $\mathbb{R}^n$ and in Hilbert space, and likely in Banach space. I don't know about your very general setting.

$\endgroup$
1
  • $\begingroup$ Thank you very much! The condition seems to be valid in a Hausdorff l.c.s. $\endgroup$
    – Mikhail
    Sep 12 '21 at 9:23

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.