1norm

The 1-norm, also called the L1 norm or Manhattan norm, of a vector x in R^n is defined as ||x||_1 = sum_{i=1}^n |x_i|. It is one of the standard p-norms with p = 1 and is widely used in mathematics and applied fields.

Properties and characteristics: The 1-norm satisfies the norm axioms: non-negativity, definiteness, scalability, and the triangle inequality. It is convex and piecewise linear, with non-differentiability at coordinates where x_i = 0. The unit ball {x : ||x||_1 <= 1} is a cross-polytope; in two dimensions it appears as a diamond shape, with higher-dimensional analogs.

Dual norm and relationships: The dual norm of the L1 norm is the L∞ norm, defined by ||z||_∞ = max_i |z_i|. This duality links L1 to maximum-coordinate measurements and has implications in optimization and sensitivity analysis.

Applications and optimization: The L1 norm is widely used as a regularizer in optimization problems to promote sparsity, as in Lasso regression and sparse signal recovery. In such settings, minimizing ||x||_1 under data-fitting constraints tends to produce solutions with many zero components. The subgradient of ||x||_1 is sign(x_i) for x_i ≠ 0 and can take any value in [-1, 1] when x_i = 0, reflecting its non-smooth nature.

Distances and variants: The L1 distance between vectors x and y is ||x − y||_1. Variants include weighted L1 norms and group L1 norms, which introduce structure or feature-specific penalties.

Example: For x = (3, -2, 0), ||x||_1 = 3 + 2 + 0 = 5.