Let X be a separable, normed space, t a positive real number. Denote by \(F_{t,X}(p)\) the greatest lower bound of the probabilities \(P(\| \xi +\eta \| \geq ta)\), where (\(\xi\),\(\eta\)) means an arbitrary couple of L i.i.d. X-valued random vectors with a symmetric distribution such that \(P(\| \xi \| \geq a)=p\). The function \(F_{t,X}\) is found for \(t\leq 1\) in Theorem 1. In Theorem 2, the author has shown that for \(t>1\), \(F_{t,X}\) can be expressed through some geometrical constants introduced by him.