Alternative Geometries

Question

In our world, the distance between two points (in 2d) is defined as $\sqrt{(\Delta x)^2 + (\Delta y)^2}$. Suppose that in an alternative geometry, it was defined as $\sqrt[p]{|\Delta x|^p + |\Delta y|^p}$. How would the shapes of geodesics change? How would angles be defined? Is there a way of visualizing such a geometry as a smooth surface? Is there an intuitive way of thinking about it? Could someone provide a reference to the subject?

Answer 1

Here are some thoughts on the subject.

Using such a metric locally turns the Euclidean space into a Finsler manifold instead of a Riemannian manifold. If you are not familiar with differential geometry, you might want to ignore this. You should also check the link Noah Schweber gave in a comment: Wikipedia tells about $\ell^p$-norms in $\mathbb R^n$.

One striking difference is that there would be preferred directions, the coordinate axes. The usual metric is rotationally invariant and we can set coordinates quite freely. For the norm you propose this is no longer the case unless $p=2$. Due to this lack of symmetry balls will no longer be round, for example, and a rotated ball is typically not a ball.

One way to rephrase is that if rotations are defined to be linear isometries, the only rotations are obtained by permuting and flipping the coordinate axes. Rotations only come in steps of 90 degrees and there are no small rotations. (The rotation group is finite and generated by 90 degree rotations between pairs of coordinate directions.)