If $\Omega$ is a bounded open set in $R^n$, $\Omega_j\subset\Omega$, and $|\Omega_j|\geq\epsilon$, which $\epsilon$ is a constant. Can we say there is a subsequent $\Omega_{j_k}$ of $\Omega_j$, such that $|\bigcap_k\Omega_{j_k}|>0$ ? We know this is true when $\Omega_j$ are all balls. So, considering what assumption， this is true?

If $\Omega$ is a bounded open set in $R^n$, $\Omega_j\subset\Omega$, and $|\Omega_j|\geq\epsilon$, which $\epsilon$ is a constant. Can we say there is a subsequence $\Omega_{j_k}$ of $\Omega_j$, such that $|\bigcap_k\Omega_{j_k}|>0$ ? We know this is true when $\Omega_j$ are all balls. So, considering what assumption， this is true?

If $\Omega$ is a bounded open set in $R^n$, $\Omega_j\subset\Omega$, and $|\Omega_j|\geq\epsilon$, can we say there is a subsequent $\Omega_{j_k}$ of $\Omega_j$, such that $|\bigcap_k\Omega_{j_k}|>0$ ? We know this is true when $\Omega_j$ are all balls. So, considering what assumption， this is true?

If $\Omega$ is a bounded open set in $R^n$, $\Omega_j\subset\Omega$, and $|\Omega_j|\geq\epsilon$, which $\epsilon$ is a constant. Can we say there is a subsequent $\Omega_{j_k}$ of $\Omega_j$, such that $|\bigcap_k\Omega_{j_k}|>0$ ? We know this is true when $\Omega_j$ are all balls. So, considering what assumption， this is true?