Assuming both players react instantaneously, and space is modelled as continuous (not discrete), I think the following strategy always wins for the lion.

    1. Lion runs to centre of ring
    2. Lion moves toward tamer, but always staying directly between tamer and centre.

(2) is always possible, as the arc the lion has to move around to keep between tamer and centre is always shorter than any arc the tamer can move along since the lion is closer in. As a corollory, the lion can always get closer to the tamer unless the tamer moves direclty away from the centre.

Eventually the tamer reaches the edge of the ring and can no longer move directly away. Then the lion can continue to move outward, always between the tamer and centre, until they meet, and the lion eats.