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I'd like to know if there exists an established name for a set $T \subset \mathbb{R}^n$ with the following property:

For every $x \in T$ there exists a convex set $C \subset T$ such that $x \in C$ and $\lambda(C) > 0$
where $\lambda$ is the Lebesgue measure.

Intuitively, what I am trying to describe is a set that "has volume everywhere", such that a point $x \in T$ can be moved a small distance in any of $n$ independent directions and still remain inside $T$.

janismac
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  • Isn't that condition equivalent to being everywhere dense? – Asinomás Apr 26 '17 at 15:34
  • nevermind, not quite. – Asinomás Apr 26 '17 at 15:34
  • If you replace "convex" with "open", then I think your statement is basically the same as requiring that the measure space induced on $T$ has support the whole of $T$. A supported point in a measure space is a point in which every open neighbourhood of that point has positive measure; the support of a measure is all the supported points. – Joppy Apr 29 '17 at 14:19

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