n1ball

n1ball, also written as the n1-ball, is the unit ball of the L1 norm in n-dimensional space. It is defined as B1^n = { x ∈ R^n : ||x||1 ≤ 1 } where ||x||1 = sum_i |x_i|. This makes n1ball the set of all vectors whose total absolute value does not exceed one.

Geometrically, B1^n is the cross-polytope in n dimensions. It is a convex, compact, and centrally symmetric polytope with 2n vertices located at the coordinate directions ±e_i (the standard basis vectors). Its boundary consists of points where sum_i |x_i| = 1, and in two dimensions it appears as a diamond; in three dimensions it is an octahedron.

The unit ball for the dual norm, the infinity norm, is the hypercube B∞^n = { y ∈ R^n : max_i |y_i| ≤ 1 }. In the sense of convex duality, B∞^n is the polar of B1^n. The L1 ball also has a known volume, equal to 2^n / n!, reflecting its growing complexity as the dimension increases. In optimization, B1^n is commonly used as a constraint or a regularizer.

Notation varies; some literature refers to it literally as the n1-ball, while many mathematical texts denote it as B1^n to emphasize the L1 norm. Its prominence comes from properties that promote sparsity: optimization problems with L1 penalties tend to produce solutions with many zero components, a feature exploited in sparse recovery and compressed sensing.

Overall, the n1-ball is a fundamental geometric object in higher-dimensional analysis, linking norms, duality, and convex optimization in a compact, polyhedral form.