Brouwer's fixed point theorem only applies to compact convex sets. Infinite lines don't work, as they are not compact. Similarly a circle would not work as it

Brouwer's fixed point theorem only applies to compact convex sets. Infinite lines don't work, as they are not compact. Similarly a circle would not work as it is not convex (you're close with your example, you just need to glue together the endpoints to turn it into a circle and make the map continuous).