mathematics

Minkowski's first inequality for convex bodies

Minkowski's first inequality for convex bodies is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Minkowski's first inequality for convex bodies rather than just read about it. In short: In mathematics, Minkowski's first inequality for convex bodies is a geometrical result due to the German mathematician Hermann Minkowski. The inequality is closely related to the Brunn–Minkowski inequality and the isoperimetric inequality.

Key takeaways

Minkowski's first inequality for convex bodies belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Minkowski's first inequality for convex bodies to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Minkowski's first inequality for convex bodies from memory before moving on to harder problems.

Reference excerpt

In mathematics, Minkowski's first inequality for convex bodies is a geometrical result due to the German mathematician Hermann Minkowski. The inequality is closely related to the Brunn–Minkowski inequality and the isoperimetric inequality.

Statement of the inequality Let K and L be two n-dimensional convex bodies in n-dimensional Euclidean space Rn. Define a quantity V1(K, L) by

n V 1 ( K , L ) = lim ε ↓ 0 V ( K + ε L ) − V ( K ) ε , {\displaystyle nV_{1}(K,L)=\lim _{\varepsilon \downarrow 0}{\frac {V(K+\varepsilon L)-V(K)}{\varepsilon }},}

where V denotes the n-dimensional Lebesgue measure and + denotes the Minkowski sum. Then

V 1 ( K , L ) ≥ V ( K ) ( n − 1 ) / n V ( L ) 1 / n , {\displaystyle V_{1}(K,L)\geq V(K)^{(n-1)/n}V(L)^{1/n},}

with equality if and only if K and L are homothetic, i.e. are equal up to translation and dilation.

Remarks V1 is just one example of a class of quantities known as mixed volumes. If L is the n-dimensional unit ball B, then n V1(K, B) is the (n − 1)-dimensional surface measure of K, denoted S(K).

Connection to other inequalities

The Brunn–Minkowski inequality One can show that the Brunn–Minkowski inequality for convex bodies in Rn implies Minkowski's first inequality for convex bodies in Rn, and that equality in the Brunn–Minkowski inequality implies equality in Minkowski's first inequality.

The isoperimetric inequality By taking L = B, the n-dimensional unit ball, in Minkowski's first inequality for convex bodies, one obtains the isoperimetric inequality for convex bodies in Rn: if K is a convex body in Rn, then

( V ( K ) V ( B ) ) 1 / n ≤ ( S ( K ) S ( B ) ) 1 / ( n − 1 ) , {\displaystyle \left({\frac {V(K)}{V(B)}}\right)^{1/n}\leq \left({\frac {S(K)}{S(B)}}\right)^{1/(n-1)},}

with equality if and only if K is a ball of some radius.

References Gardner, Richard J. (2002). "The Brunn–Minkowski inequality". Bull. Amer. Math. Soc. (N.S.). 39 (3): 355–405 (electronic). doi:10.1090/S0273-0979-02-00941-2.

Worked examples

Example 1 — a first encounter with Minkowski's first inequality for convex bodies

Start with the simplest possible case. Write down what Minkowski's first inequality for convex bodies claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Minkowski's first inequality for convex bodies before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Minkowski's first inequality for convex bodies ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Minkowski's first inequality for convex bodies

In research: Minkowski's first inequality for convex bodies appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Minkowski's first inequality for convex bodies in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Minkowski's first inequality for convex bodies is common in secondary-school and first-year university syllabi. It links to neighbouring topics Calculus of variations, Geometric inequalities, Metric geometry stubs, so understanding it makes those chapters shorter.
In everyday life: Look for Minkowski's first inequality for convex bodies outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Minkowski's first inequality for convex bodies in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Minkowski's first inequality for convex bodies means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Minkowski's first inequality for convex bodies out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Minkowski's first inequality for convex bodies in simple terms?

Why does Minkowski's first inequality for convex bodies matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Minkowski's first inequality for convex bodies?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Minkowski's first inequality for convex bodies.