all right, let’s be clear so we all know what we are talking about:
Lightweight shoes, just like racing flats can be anything. High arch, high heeled, narrow fit, drops of 4mm or more, etc. It can be anything from 4 ounces to 10 ounces, anything, depending your needs (and size), and they usually have some degree of cushioning. They are just lighter.
Minimalist shoes are always lightweight but also they are always zero drop, that is, no height difference between the front and the back of the shoe (most shoes, even lightweight ones, have a heel). Minimalist shoes need to be very flexible but also have a very thin and dense sole, 4 or 3 mm or under. They need a wide toebox to allow the foot to splay out when collapsing on contact and they need to feel like you’re almost barefoot when you need some protection.
Barefoot shoes: there is no such thing. What part of bare feet are we not clear about?
(no, not even running sandals are barefoot anything!)
More on Running Biomechanics
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