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Today Kyle Korver wears Nike, but that hasn't always been the case. Korver used to sport Converse, and even had a sneaker deal with the brand. I cannot think of a reason why a basketball player would choose to wear Converse, but getting paid can motivate people to do the unthinkable. Kyle wore Converse sneakers during his time on the Jazz, and Bulls. He probably doesn't sport Converse kicks today due to their lack of innovation. Converse has basically been selling the same shoe design since their establishment in 1908.
If you ever wanted to hear Kyle talk about Converse sneakers; here is a video from 2010:
Currently Kyle Korver wears Nike shoes, and this is a huge upgrade from Converse. This season he has been wearing the Kobe X. In addition to the Kobe's, last year he wore the Nike Run The One. During the 2015 All Star Week, Korver gained a lot of attention for his player exclusive kicks.