Admittedly my knowledge of canes is only limited to being on the receiving end of a beatdown (an incident at the senior center, I can’t talk about it- too embarrassing) but it does make sense that it would be hard to use a cane in icy conditions. Unless you had a Duro-Med Retractable Ice Tip Cane. Just push the button and the ice tip extends out from the bottom allowing the cane to get a good grip in the ice giving you an added measure of safety outdoors. Plus it adds an extra punch for cane-related beatings.
Sminiker LED Folding Walking Cane with Carrying Bag for Old Gentleman or Lady Aluminum Alloy Foldable Cane with Light