What good would an ice axe be if your running, your not able to use it as a walking support in your leading hand because your running, and unless you know how to self arrest then it would be practically useless, and running with an axe seems to me a touch dangerous.
Personally if you feel the need to have an ice axe then you should most likely not run but just walk, and stick to hills that if you fall theres no danger of sliding your way down the hill, unless am missing something.
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