  We show that around any $m$-partite product state $\rho_{\rm
prod}=\rho_1\otimes...\otimes\rho_m$ of full rank (that is ${\rm det}(\rho_{\rm
prod})\neq 0)$, there exists a finite-sized closed ball of separable states
centered around $\rho_{\rm prod}$ whose radius is $\beta:=2^{1-m/2}\lambda_{\rm
min}(\rho_{\rm prod})$. Here, $\lambda_{\rm min}(\rho_{\rm prod})$ is the
smallest eigenvalue of $\rho_{\rm prod}$. We are assuming that the total
Hilbert space is finite dimensional and we use the notion of distance induced
by the Frobenius norm. Applying a scaling relation, we also give a new and
simple sufficient criterion for multipartite separability based on trace: ${\rm
Tr}[\rho\rho_{\rm prod}]^2/{\rm Tr}[\rho^2]\geq {\rm Tr}[\rho_{\rm
prod}^2]-\beta^2$. Using the separable balls around the full-rank product
states, we discuss the existence and possible sizes of separable balls around
any multipartite separable states, which are important features for the set of
all separable states. We discuss the implication of these separable balls on
entanglement dynamics.
