Minkowski Inequality

Minkowski inequality (also known as Brunn Minkowski inequality) states that if two functions ‘f’ and ‘g’ and their sum (f + g) is measurable, then for 1 ≤ p < ∞, 

||f + g||_p ≤ ||f||_p + ||g||_p which implies ${\left( \int \left| f+g\right| ^{p}\right) ^{\dfrac{1}{p}}\leq \left( \int \left| f\right| ^{p}\right) ^{\dfrac{1}{p}}+\left( \int \left| g\right| ^{p}\right) ^{\dfrac{1}{p}}}$

Mathematically, 

If f, g Є L^p and (f + g) Є L^p, then for 1 ≤ p < ∞,

||f + g||_p ≤ ||f||_p + ||g||_p 

⇒ ${\left( \int \left| f+g\right| ^{p}\right) ^{\dfrac{1}{p}}\leq \left( \int \left| f\right| ^{p}\right) ^{\dfrac{1}{p}}+\left( \int \left| g\right| ^{p}\right) ^{\dfrac{1}{p}}}$

This is the generalized Minkowski inequality for integrals.

In the field of mathematical analysis, the Minkowski inequality confirms that L^p spaces qualify as normed vector spaces.

Proof

Firstly, we will verify that the inequality holds for p = 1, and then for 1 < p < ∞

When p = 1,

||f + g|| = ${\int \left| f+g\right|}$ ≤ ${\int \left| f\right| +\int \left| g\right|}$ = ||f||₁ + ||g||₁ 

When p > 1,

|f + g|^p = (|f + g|) ⋅ (|f + g|)^{p – 1} ≤ (|f| + |g|) ⋅ (|f + g|)^{p – 1} = |f| ⋅ (|f + g|)^{p – 1} + |g| ⋅ (|f + g|)^{p – 1} 

⇒ |f + g|^p ≤ |f| ⋅ (|f + g|)^{p – 1} + |g| ⋅ (|f + g|)^{p – 1} …..(i)

From Hölder’s inequality, we know

${\dfrac{1}{p}+\dfrac{1}{q}=1}$

⇒ ${q=\dfrac{p}{p-1}}$

⇒ ${p=q\left( p-1\right)}$

Now, |f + g|^{q(p – 1)} = |f + g|^p

⇒ ${\int \left| f+g\right| ^{q\left( p-1\right) }=\int \left| f+g\right| ^{p}}$ …..(ii)

By definition of L^p space, R.H.S. of (ii) is finite, and thus, L.H.S. of (ii) is also finite.

⇒ ${\int \left| f+g\right| \left( p-1\right)}$ Є L^q

Now, by integrating (i), we get

${\int \left| f+g\right| ^{p}\leq \int \left| f\right| \cdot \left| f+g\right| ^{p-1}+\int \left| g\right| \cdot \left| f+g\right| ^{p-1}}$

Using Hölder’s inequality in R.H.S, we get

${\int \left| f+g\right| ^{p}\leq \left\| f\right\| _{p}\cdot \left\| f+g\right\| _{q}^{p-1}+\left\| g\right\| _{p}\cdot \left\| f+g\right\| _{q}^{p-1}}$

⇒ ${\left\| f+g\right\| ^{p}\leq \left( \left\| f\right\| _{p}+\left\| g\right\| _{p}\right) \cdot \left\| f+g\right\| _{q}^{p-1}}$ …..(iii)

If ${\left\| f+g\right\| _{q}^{p-1}}$ = 0, then from (iii), ${\int \left| f+g\right| ^{p}=0}$

⇒ ${\left\| f+g\right\| ^{p}=0=\left\| f+g\right\| _{q}^{p-1}}$

If possible let ${\left\| f+g\right\| _{q}^{p-1}}$ ≠ 0

Its value of integration in (iii) is ${\int \left( \left| f+g\right| ^{p-1}\right) ^{q}}$

Now, dividing the inequality (iii) by ${\left( \int \left| f+g\right| ^{p}\right) ^{\dfrac{1}{q}}}$, we get

${\dfrac{\int \left| f+g\right| ^{p}}{\left( \int \left| f+g\right| ^{p}\right) ^{\dfrac{1}{q}}}\leq \dfrac{\left( \left\| f\right\| _{p}+\left\| g\right\| _{p}\right) \left( \int \left| f+g\right| ^{\left( p-1\right) q}\right) ^{\dfrac{1}{q}}}{\left( \int \left| f+g\right| ^{p}\right) ^{\dfrac{1}{q}}}}$

⇒ ${\dfrac{\int \left| f+g\right| ^{p}}{\left( \int \left| f+g\right| ^{p}\right) ^{\dfrac{1}{q}}}\leq \left( \left\| f\right\| _{p}+\left\| g\right\| _{p}\right)}$ (Since q(p – 1) = p)

⇒ ${\left( \int \left| f+g\right| ^{p}\right) ^{1-\dfrac{1}{q}}\leq \left\| f\right\| _{p}+\left\| g\right\| _{p}}$ …..(iv)

Since ${\dfrac{1}{p}+\dfrac{1}{q}=1}$, which means ${1-\dfrac{1}{q}=\dfrac{1}{p}}$

Thus, from (iv), we get ${\left( \int | f+g\right) ^{\dfrac{1}{p}}\leq \left\| f\right\| _{p}+\left\| g\right\| _{p}}$

⇒ ${\left\| f+g\right\| _{p}\leq \left\| f\right\| _{p}+\left\| g\right\| _{p}}$

Thus, Minkowski’s inequality is proved.

Summation Form

When expressed in the summation form, it is written as

If x, y Є L^p and (x + y) Є L^p, then for 1 ≤ p < ∞,

${\left( \sum ^{\infty }_{i=1}\left| x_{i}+y_{i}\right| ^{p}\right) ^{\dfrac{1}{p}}\leq \left( \sum ^{\infty }_{i=1}\left| x_{i}\right| ^{p}\right) ^{\dfrac{1}{p}}+\left( \sum ^{\infty }_{i=1}\left| y_{i}\right| ^{p}\right) ^{\dfrac{1}{p}}}$

Proof

When p = 1,

The inequality follows from the triangular inequality of real numbers.

When p > 1,

${\sum ^{\infty }_{i=1}\left| x_{i}+y_{i}\right| ^{p}=\sum ^{\infty }_{i=1}\left| x_{i}+y_{i}\right| \cdot \left| x_{i}+y_{i}\right| ^{p-1}\leq \sum ^{\infty }_{i=1}\left( \left| x_{i}\right| +\left| y_{i}\right| \right) \cdot \left| x_{i}+y_{i}\right| ^{p-1}}$

⇒ ${\sum ^{\infty }_{i=1}\left| x_{i}+y_{i}\right| ^{p}\leq \sum ^{\infty }_{i=1}\left| x_{i}\right| \cdot \left| x_{i}+y_{i}\right| ^{p-1}+\sum ^{\infty }_{i=1}\left| y_{i}\right| \cdot \left| x_{i}+y_{i}\right| ^{p-1}}$ …..(i)

From Hölder’s inequality, we know

${\dfrac{1}{p}+\dfrac{1}{q}=1}$

⇒ ${q=\dfrac{p}{p-1}}$

⇒ ${p=q\left( p-1\right)}$

Using Hölder’s inequality in R.H.S. of (i), we get

${\sum ^{\infty }_{i=1}\left| x_{i}+y_{i}\right| ^{p}\leq \left( \sum ^{\infty }_{i=1}\left| x_{i}\right| ^{p}\right) ^{\dfrac{1}{p}}\left( \sum ^{\infty }_{i=1}\left| x_{i}+y_{i}\right| ^{q\left( p-1\right) }\right) ^{\dfrac{1}{q}}+\left( \sum ^{\infty }_{i=1}\left| y_{i}\right| ^{p}\right) ^{\dfrac{1}{p}}\left( \sum ^{\infty }_{i=1}\left| x_{i}+y_{i}\right| ^{q\left( p-1\right) }\right) ^{\dfrac{1}{q}}}$ …..(ii)

Now, dividing the inequality (ii) by ${\left( \sum ^{\infty }_{i=1}\left| x_{i}+y_{i}\right| ^{p}\right) ^{\dfrac{1}{q}}}$, we get

${\dfrac{\sum ^{\infty }_{i=1}\left| x_{i}+y_{i}\right| ^{p}}{\left( \sum ^{\infty }_{i=1}\left| x_{i}+y_{i}\right| ^{p}\right) ^{\dfrac{1}{q}}}\leq \dfrac{\left( \sum ^{\infty }_{i=1}\left| x_{i}\right| ^{p}\right) ^{\dfrac{1}{p}}\left( \sum ^{\infty }_{i=1}\left| x_{i}+y_{i}\right| ^{q\left( p-1\right) }\right) ^{\dfrac{1}{q}}}{\left( \sum _{i=1}\left| x_{i}+y_{i}\right| \right) ^{p}\dfrac{1}{q}}+\dfrac{\left( \sum ^{\infty }_{i=1}\left| y_{i}\right| ^{p}\right) \dfrac{1}{p}\left( \sum ^{\infty }_{i=1}\left| x_{i}+y_{i}\right| ^{q\left( p-1\right) }\right) ^{\dfrac{1}{q}}}{\left( \sum _{i=1}\left| x_{i}+y_{i}\right| \right) ^{p}\dfrac{1}{q}}}$

⇒ ${\dfrac{\sum ^{\infty }_{i=1}\left| x_{i}+y_{i}\right| ^{p}}{\left( \sum ^{\infty }_{i=1}\left| x_{i}+y_{i}\right| ^{p}\right) ^{\dfrac{1}{q}}}\leq \left( \sum ^{\infty }_{i=1}\left| x_{i}\right| ^{p}\right) ^{\dfrac{1}{p}}+\left( \sum ^{\infty }_{i=1}\left| y_{i}\right| ^{p}\right) ^{\dfrac{1}{p}}}$ (Since q(p – 1) = p)

⇒ ${\left( \sum ^{\infty }_{i=1}\left| x_{i}+y_{i}\right| ^{p}\right) ^{1-\dfrac{1}{q}}\leq \left( \sum ^{\infty }_{i=1}\left| x_{i}\right| ^{p}\right) ^{\dfrac{1}{p}}+\left( \sum ^{\infty }_{i=1}\left| y_{i}\right| ^{p}\right) ^{\dfrac{1}{p}}}$ …..(iii)

Since ${\dfrac{1}{p}+\dfrac{1}{q}=1}$, which means ${1-\dfrac{1}{q}=\dfrac{1}{p}}$

Thus, from (iii), we get

${\left( \sum ^{\infty }_{i=1}\left| x_{i}+y_{i}\right| ^{p}\right) ^{\dfrac{1}{p}}\leq \left( \sum ^{\infty }_{i=1}\left| x_{i}\right| ^{p}\right) ^{\dfrac{1}{p}}+\left( \sum ^{\infty }_{i=1}\left| y_{i}\right| ^{p}\right) ^{\dfrac{1}{p}}}$

Thus, Minkowski’s inequality is proved.