There are many brands out there to choose from when it comes to
cross training shoes. A famously known brand who make cross
training shoes are Nike. They sell both online and in sports
shops.
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Shoes are protection for the feet. Any form of training that
involves using the feet for support should be accompanied by the
use of a good pair of shoes. Different training shoes have been
developed over the years. You might need to decide what the main
focus of your training is to help guide you toward the correct type
of shoe to use.
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The most common and highly recommended shoe companies that make
cross training shoes are New Balance, Asics, Nike, Ryka, and also
Converse cross training shoes.
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Good women training shoes are made by nike. I would look into
the lunarglide series. Theay are top of the line running shoes and
are great for whatever stage of training you are at. They are worn
by many pro athletes.