I start just over 12.5 miles from the south pole, let’s say 12.54 miles.

After walking south I am 2.54 miles away from the pole.

Walking east I am still 2.54 miles from the pole, but have travelled around 225.7 degrees around it.

Heading north, directly away from the pole, I end up 12.54 miles from the pole.

Finally heading west I reverse the direction around the pole, but this time by about 45.7 degrees, ending up roughly opposite where I started, but around 25.08 miles from it.

In fact if I’d started a little more than 12.54 miles from the south pole my final position would be even further away, but would depend on the curvature of the earth, but even assuming local flatness it’s no more than about 25.23 miles.

The exact maths of ending up opposite where you started is that you would start 10+10*(sqrt(pi*(4+pi))-pi)/(2*pi)=12.53863… miles away from the South Pole. The distance between start and end are then just double that.

The overall maximisation is beyond my abilities.