In topological vector space, intuitively a balanced set is a set which doesn't have a hole. Suppose $V$ is a convex set containing origin, can anyone give a counterexample which is not balanced?
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2The interval $(-1,2)$ is an example in $\Bbb R$. – David C. Ullrich May 18 '16 at 22:29
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In $\mathbb{C}$, take a square with the origin as center.
mrf
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(or more generally, take any convex set containing $0$ that is not a disc) – mrf May 19 '16 at 13:00