Surprising shapes in space

The previous article showed that, even in the plane, the notion of a "ball" gives rise to some surprising shapes. What happens in space?

Balls that grow… ---------------------------

Let p be a strictly positive real number. In ordinary space—that is, ℝ³ equipped with an orthonormal coordinate system—we define the norm N*_p* of the vector in ℝ³ with coordinates (x, y, z) by ${\text{N}}_p(x,y,z)=(\mathrm{\mid }x{\mathrm{\mid }}^p+\mathrm{\mid }y{\mathrm{\mid }}^p+\mathrm{\mid }z{\mathrm{\mid }}^p{)}^{1\mathrm{/}p},\backslash text\{N\}\_p\; (x,\; y,\; z)\; =\; (\; |\; x\; |^p\; +\; |\; y\; |^p\; +\; |\; z\; |^p\; )\; ^\{1/p\},$ just as for vectors in the plane (see page 36). This then allows us to define the distance associated with N*_p*.

As in the plane, N*_p is a norm only when p ≥ 1. When p = 2, we recover Euclidean distance, for which the shortest path between two points is a straight line. When p = 1, the distance from the origin O(0, 0, 0) to the point with coordinates (x, y, z*) is attained by a path consisting of three line segments parallel to the axes, generalizing the "Manhattan" distance.

The unit ball B*_p(O, 1) in space, or simply B_p, is defined in the same way as in the plane; its boundary* is the surface with equation $\mathrm{\mid }x{\mathrm{\mid }}^p+\mathrm{\mid }y{\mathrm{\mid }}^p+\mathrm{\mid }z{\mathrm{\mid }}^p=1.|\; x\; |^p\; +\; |\; y\; |^p\; +\; |\; z\; |^p\; =\; 1.$