Yes, I also have the zig-zag in mind. However, as a point, the lion will never catch him. The lion has to move further to the side than the tamer, so the tamer

Yes, I also have the zig-zag in mind. However, as a point, the lion will never catch him. The lion has to move further to the side than the tamer, so the tamer switches directions after each "step". This works, because within a circle the tamer with his back to the wall has to move a little towards the lion while moving sideways. The lion moves a little bit more to the side a little bit less towards the tamer. Over time the lion comes closer and the steps become smaller. When it gets really close however, the round wall looks increasingly like a flat wall. Lion and tamer end up just moving sideways without ever touching? We need some more math for this.