Balls in the plane

In the plane, a ball is the set of points whose distance (in the usual Euclidean sense) from a given center is less than or equal to a constant. This familiar ball is not really a "ball" as an enthusiast of jeu provençal, a pétanque player, or a raffa volo champion would understand the word, but a disk. Moreover, balls defined using arbitrary distances can have far more outlandish shapes (as the article "Some surprising distances" amply demonstrates). But let's stay in the familiar plane and focus on distances induced by a "norm" (see box), with the distance between two points A and B in the plane defined as the norm of the vector $\overrightarrow{AB}\mathrm{.}\backslash overrightarrow\{AB\}.$

In search of generalization --------------------------

In the usual plane—that is, ℝ² equipped with an orthonormal coordinate system—the Euclidean norm N₂ is defined by ${\text{N}}_2(x,y)=\sqrt{x^2+y^2}=(x^2+y^2{)}^{1\mathrm{/}2},\backslash text\{N\}\_2\; (x,\; y)\; =\; \backslash sqrt\{x^2\; +\; y^2\}\; =\; (x^2\; +y^2)^\{1/2\},$ which represents the "straight-line" distance from the origin (0, 0) to the point with coordinates (x, y).

Seeking to generalize, mathematicians replaced the number 2 with a strictly positive real number p and defined N*_p by setting N_p (x, y) = (x ^p +y ^p *)^1/*^p, an expression that remains meaningful for every p > 0 and all real numbers x and y. We can then verify that N_p is indeed a norm… but only when p* ≥ 1.