Minkowski's Inequality

The statement and what it means

Let (X, Σ, μ) be a measure space and 1 ≤ p ≤ ∞. Define the L^p norm by

‖f‖_p = (∫_X |f|^p dμ)^(1/p)             for 1 ≤ p < ∞
‖f‖_∞ = ess sup |f|                        for p = ∞

Minkowski's inequality says that for any measurable f, g:

‖f + g‖_p ≤ ‖f‖_p + ‖g‖_p

Equivalently, on sequence spaces (counting measure on ℕ):

(Σ |a_k + b_k|^p)^(1/p) ≤ (Σ |a_k|^p)^(1/p) + (Σ |b_k|^p)^(1/p)

And for vectors in ℝⁿ (counting measure on {1, …, n}), the same formula. The case p = 2 is the familiar Euclidean triangle inequality ‖a + b‖ ≤ ‖a‖ + ‖b‖.

The geometric content is simple: the diagonal of a parallelogram with sides f and g is no longer than the sum of the two sides. Minkowski's contribution is to make this statement quantitatively precise for any exponent p ≥ 1, in any dimension, on any measure space.

Proof sketch — Hölder is the engine

For p = 1 and p = ∞ the proof is direct. For p = 1: |f + g| ≤ |f| + |g| pointwise, integrate. For p = ∞: take essential sup of both sides.

For 1 < p < ∞, write |f + g|^p = |f + g| · |f + g|^(p−1), apply the pointwise triangle inequality |f + g| ≤ |f| + |g|, and split:

∫ |f + g|^p ≤ ∫ |f| · |f + g|^(p−1) + ∫ |g| · |f + g|^(p−1)

Now apply Hölder's inequality to each term with conjugate exponent q = p / (p − 1) (so 1/p + 1/q = 1):

∫ |f| · |f + g|^(p−1) ≤ ‖f‖_p · ‖|f + g|^(p−1)‖_q
                       = ‖f‖_p · (∫ |f + g|^p)^((p−1)/p)
                       = ‖f‖_p · ‖f + g‖_p^(p−1)

An identical bound holds for the g term. Summing and dividing both sides by ‖f + g‖_p^(p−1) gives:

‖f + g‖_p = ‖f + g‖_p^p / ‖f + g‖_p^(p−1) ≤ ‖f‖_p + ‖g‖_p     □

The chain is short. Hölder is the only inequality used; everything else is algebra. Minkowski rests on Hölder, which rests on Young's inequality, which rests on convexity of log. The hierarchy is tight.

Numerical examples

Example 1 (Euclidean, p = 2):
  a = (3, 4),   b = (0, 5)
  a + b = (3, 9)
  ‖a‖₂ = 5,  ‖b‖₂ = 5,  ‖a + b‖₂ = √90 ≈ 9.487
  ‖a‖₂ + ‖b‖₂ = 10
  9.487 ≤ 10            ✓

Example 2 (Manhattan, p = 1):
  a = (3, 4),   b = (0, 5)
  a + b = (3, 9)
  ‖a‖₁ = 7,  ‖b‖₁ = 5,  ‖a + b‖₁ = 12
  ‖a‖₁ + ‖b‖₁ = 12
  12 ≤ 12               EQUALITY (a, b have same-sign coordinates)

Example 3 (Chebyshev, p = ∞):
  a = (3, 4),   b = (0, 5)
  a + b = (3, 9)
  ‖a‖_∞ = 4,  ‖b‖_∞ = 5,  ‖a + b‖_∞ = 9
  ‖a‖_∞ + ‖b‖_∞ = 9
  9 ≤ 9                 EQUALITY (max coordinate in same slot)

Example 4 (p = 3):
  a = (1, 2),  b = (2, 1)
  a + b = (3, 3)
  ‖a‖₃ = (1+8)^(1/3) = 9^(1/3) ≈ 2.080
  ‖b‖₃ ≈ 2.080
  ‖a+b‖₃ = (27+27)^(1/3) = 54^(1/3) ≈ 3.780
  ‖a‖₃ + ‖b‖₃ ≈ 4.160
  3.780 ≤ 4.160         ✓

Example 5 (equality at p = 2):
  a = (3, 4),   b = (6, 8)   (b = 2a)
  ‖a‖₂ = 5,  ‖b‖₂ = 10,  ‖a + b‖₂ = 15
  ‖a‖₂ + ‖b‖₂ = 15
  15 = 15               EQUALITY (a, b proportional, same direction)

The geometric pattern is consistent: equality in the L² triangle inequality holds when the vectors are non-negatively parallel; for p = 1 and p = ∞ equality is easier to achieve.

Variants and generalizations

• Triangle inequality on ℝⁿ. The classical ‖a + b‖₂ ≤ ‖a‖₂ + ‖b‖₂ is Minkowski with p = 2 and counting measure on {1, …, n}.
• Sequences (ℓ^p). (Σ|aₖ + bₖ|^p)^(1/p) ≤ (Σ|aₖ|^p)^(1/p) + (Σ|bₖ|^p)^(1/p). Makes ℓ^p a normed (and hence Banach) space.
• Functions (L^p). (∫|f + g|^p)^(1/p) ≤ (∫|f|^p)^(1/p) + (∫|g|^p)^(1/p). Makes L^p a normed space and (after completion) a Banach space.
• Minkowski's integral inequality. ‖∫ f(·, y) dy‖_p ≤ ∫ ‖f(·, y)‖_p dy. Continuous-parameter version of triangle inequality. Used for bounding integral operators.
• Reverse Minkowski (0 < p < 1). The inequality reverses: ‖f + g‖_p ≥ ‖f‖_p + ‖g‖_p for non-negative f, g. So ‖·‖_p is not a norm; the natural translation-invariant metric is d(f, g) = ‖f − g‖_p^p, which is subadditive (an F-space metric).
• Minkowski-functional / gauge. For a convex symmetric set A containing the origin, p_A(x) = inf{t > 0 : x ∈ tA} is the Minkowski gauge — subadditive by the same convexity argument. Every norm is a Minkowski gauge of its unit ball.
• Weighted Minkowski. With a positive weight w(x), ‖f + g‖_{p, w} ≤ ‖f‖_{p, w} + ‖g‖_{p, w} where ‖·‖_{p, w} = (∫|·|^p w)^(1/p). Same proof.
• Anisotropic / mixed-norm Minkowski. ‖f‖_{L^p_x L^q_y} = (∫(∫|f|^q dy)^(p/q) dx)^(1/p) satisfies Minkowski in each slot separately.

Why Minkowski fails for p < 1

The proof above uses Hölder, which requires p ≥ 1 for conjugate q ≥ 1 to make sense. More fundamentally, for 0 < p < 1 the function t ↦ t^p is concave on [0, ∞), not convex; the unit ball of ‖·‖_p in ℝ² becomes a non-convex "astroid" shape rather than a convex disk. A counter-example for p = 1/2 on ℝ² with the natural definition:

a = (1, 0),  b = (0, 1)
‖a‖_{1/2} = 1,  ‖b‖_{1/2} = 1
‖a + b‖_{1/2} = (1^{1/2} + 1^{1/2})^2 = 4
4 > 2 = ‖a‖_{1/2} + ‖b‖_{1/2}              triangle inequality FAILS

So for p < 1, ‖·‖_p is not even subadditive. The remedy is to use d(f, g) = ‖f − g‖_p^p (without the outer 1/p root) as a metric — this is subadditive and translation-invariant, and makes L^p (0 < p < 1) an F-space (a complete metric vector space). It is not a Banach space because no norm exists.

When is the bound attained?

For 1 < p < ∞, ‖f + g‖_p = ‖f‖_p + ‖g‖_p holds iff there exist non-negative constants α, β not both zero with αf = βg almost everywhere. Geometrically: f and g point in the same direction. The proof traces equality through Hölder: each Hölder application requires |f|^p ∝ |f + g|^p, i.e. f ∝ f + g, i.e. f ∝ g; and the sign-condition forces both proportionality constants to be non-negative.

For p = 1: equality holds whenever f and g have the same sign almost everywhere (because |f + g| = |f| + |g| pointwise on the agreement set). Much easier to attain.

For p = ∞: equality means ess sup |f + g| = ess sup |f| + ess sup |g|, which requires the two functions to attain their essential suprema at the same place, with the same sign.

Common pitfalls

• Forgetting that p ≥ 1 is required. Minkowski fails for 0 < p < 1. The triangle inequality on the "fractional L^p" reverses.
• Equality requires f = λg, not just |f| = λ|g|. For p > 1, f and g must point the same way — opposite signs break equality even with proportional magnitudes.
• Confusing Minkowski with Hölder. Minkowski bounds a sum-norm by sum of norms. Hölder bounds a product-integral by product of norms. Different statements with different uses.
• "Minkowski applies to any norm". Triangle inequality is a defining axiom of any norm, so trivially yes — but "Minkowski" specifically refers to the L^p form. Calling the general triangle inequality "Minkowski" is sloppy.
• Forgetting that Minkowski makes L^p a normed space, not just an inner-product space. Only L² has a compatible inner product; other L^p spaces are normed but not Hilbert (no inner product, no Cauchy-Schwarz natively).
• Believing the integral inequality is "obvious". Minkowski's integral inequality ‖∫f(·,y)dy‖_p ≤ ∫‖f(·,y)‖_p dy reverses the natural order of operations; the proof is a careful application of duality (the discrete triangle inequality being the warm-up).

Where Minkowski shows up

• L^p spaces and Banach spaces. Minkowski is the triangle inequality that makes ‖·‖_p a norm. After completion, L^p is a Banach space for 1 ≤ p ≤ ∞. Half of functional analysis is L^p theory.
• Probability theory. ‖X + Y‖_p ≤ ‖X‖_p + ‖Y‖_p bounds the p-th moment of a sum by the sum of p-th moments (Minkowski for random variables). The p = 2 case: standard deviation of a sum ≤ sum of standard deviations (in L², not in general; uncorrelated stronger bound is Pythagoras).
• Functional analysis. Bounded linear operators between L^p spaces satisfy ‖Tf + Tg‖_p ≤ ‖Tf‖_p + ‖Tg‖_p by linearity and Minkowski. Combined with Hölder, Minkowski gives the L^p–L^q duality.
• PDE estimates. Splitting a solution as u = u_lin + u_nonlin and bounding ‖u‖_p ≤ ‖u_lin‖_p + ‖u_nonlin‖_p is the standard energy-method recipe.
• Image and signal processing. Decomposing a signal into low- and high-frequency parts and bounding the L^p norm of the sum by Minkowski is fundamental to wavelet and Fourier denoising bounds.
• Geometry of numbers. Minkowski's original use: counting lattice points in convex bodies. The Minkowski sum A + B of convex sets has volume bounded below by Brunn-Minkowski — a deep refinement of the triangle inequality.
• Statistics. Bias-variance decomposition ‖estimator − target‖_p ≤ ‖estimator − E[estimator]‖_p + ‖E[estimator] − target‖_p is Minkowski applied to expectation differences.
• Optimization. Convexity of the L^p norm follows from Minkowski + homogeneity. Convexity makes L^p-regularized problems tractable (LASSO is p = 1; ridge is p = 2).