Minkowski inequality

The Minkowski inequality states that the norm of a sum is at most the sum of the norms. It is the triangle inequality for finite-dimensional p-norms and Lp spaces when p ≥ 1.

Hölder's inequality: An inequality bounding the integral or sum of a product by the product of conjugate-exponent norms. Applying Hölder’s inequality to the expanded norm yields the Minkowski bound.
Lp space: A space of measurable functions whose absolute p-th powers have finite integral, with functions identified almost everywhere. Minkowski proves that the Lp expression is a norm for p ≥ 1.
Minkowski integral inequality: An integral inequality that bounds the norm of an integral of functions by the integral of their norms. It extends the norm-of-a-sum bound from discrete sums to integrals of functions.
Quasi-norm: A size function satisfying norm properties except that its triangle bound may include a fixed multiplicative constant. For 0 < p < 1, the p-expression is a quasi-norm, not a norm satisfying Minkowski’s bound.
Triangle inequality: A norm axiom stating that the size of a sum is no greater than the sum of the sizes. Minkowski is the triangle inequality specialized to Lp and p-norms.
Lp norm: For p ≥ 1, the Lp norm is the p-th root of the integral of a function’s absolute p-th power. The inequality bounds the Lp norm of a sum by the sum of the individual norms.
Minkowski inequality for sums: The finite-sum inequality bounding the p-norm of a sum of vectors by the sum of their p-norms. This is the discrete form used for vectors and sequences.
Reverse triangle inequality: A norm inequality stating that the difference between two vector lengths is at most the length of their difference. It is a consequence of the triangle inequality and gives a lower bound rather than Minkowski’s upper bound.
Conjugate exponents: Exponents p and q greater than one satisfying 1/p + 1/q = 1. The proof uses the exponent conjugate to p in Hölder’s inequality.
p-norm: A vector norm defined by the p-th root of the sum of coordinate absolute values raised to p. The finite-dimensional form bounds the p-norm of a vector sum.

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