Maximum diameter of convex cone of solutions

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I am trying to find an upper bound for the maximum distance between 2 vectors that satisfy $$x, y \in \{s|As \geq 0 \text{ and } \lvert \lvert s \rvert \rvert_2 \leq L\}$$. What is the maximum possible value of $$\lvert \lvert x - y \rvert \rvert_2$$. Do you guys knows of any upper bound of these values?