Girl and the ogre

[bonanova]

Sorry if this question appeared before, I thought I posted it once, but I searched. In any case I don't think the answer has been posted.

A girl in a boat on a circular lake is stalked by an ogre on shore who can't swim. She can out-run the ogre and escape if she can get to shore before the ogre reaches her landing point. If the boat can attain 1/pi of the ogre's speed she can win the race to shore by starting from lake's center and heading away from the ogre. The "chestnut" solution to this puzzle is that she can escape with a slower boat - one that attains only 1/[1 + pi] of the ogre's land speed.

But that's still faster than the boat needs to be. What really is the minimum boat speed that permits the girl to escape?
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[esme]

Hmm, heading straight away from the ogre at each moment seems to give the speed 1/4 which is much better than the 1/pi of keeping the initial heading. So I suspect that a combination of first using a small circle and then heading directly away from the ogre at each moment will give a better value than 1/(1 plus pi), but I dont really see a good reason why this would give the optimal value.

In any case, it is never a mistake to use the small circle, and then one should try to find the maximal guaranteed angular distance to the ogre at each radial distance.
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