mathematics

Milman's reverse Brunn–Minkowski inequality

Milman's reverse Brunn–Minkowski inequality 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 Milman's reverse Brunn–Minkowski inequality rather than just read about it. In short: In mathematics, particularly, in asymptotic convex geometry, Milman's reverse Brunn–Minkowski inequality is a result due to Vitali Milman that provides a reverse inequality to the famous Brunn–Minkowski inequality for convex bodies in n-dimensional Euclidean space Rn. Namely, it bounds the volume of the Minkowski sum of two bodies from above in terms of the volumes of the bodies.

Key takeaways

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

Reference excerpt

In mathematics, particularly, in asymptotic convex geometry, Milman's reverse Brunn–Minkowski inequality is a result due to Vitali Milman that provides a reverse inequality to the famous Brunn–Minkowski inequality for convex bodies in n-dimensional Euclidean space Rn. Namely, it bounds the volume of the Minkowski sum of two bodies from above in terms of the volumes of the bodies.

Introduction Let K and L be convex bodies in Rn. The Brunn–Minkowski inequality states that

v o l ( K + L ) 1 / n ≥ v o l ( K ) 1 / n + v o l ( L ) 1 / n , {\displaystyle \mathrm {vol} (K+L)^{1/n}\geq \mathrm {vol} (K)^{1/n}+\mathrm {vol} (L)^{1/n}~,}

where vol denotes n-dimensional Lebesgue measure and the + on the left-hand side denotes Minkowski addition. In general, no reverse bound is possible, since one can find convex bodies K and L of unit volume so that the volume of their Minkowski sum is arbitrarily large. Milman's theorem states that one can replace one of the bodies by its image under a properly chosen volume-preserving linear map so that the left-hand side of the Brunn–Minkowski inequality is bounded by a constant multiple of the right-hand side. The result is one of the main structural theorems in the local theory of Banach spaces.

Statement of the inequality There is a constant C, independent of n, such that for any two centrally symmetric convex bodies K and L in Rn, there are volume-preserving linear maps φ and ψ from Rn to itself such that for any real numbers s, t > 0

v o l ( s φ K + t ψ L ) 1 / n ≤ C ( s v o l ( φ K ) 1 / n + t v o l ( ψ L ) 1 / n ) . {\displaystyle \mathrm {vol} (s\,\varphi K+t\,\psi L)^{1/n}\leq C\left(s\,\mathrm {vol} (\varphi K)^{1/n}+t\,\mathrm {vol} (\psi L)^{1/n}\right)~.}

One of the maps may be chosen to be the identity.

Notes

References Milman, Vitali D. (1986). "Inégalité de Brunn-Minkowski inverse et applications à la théorie locale des espaces normés. [An inverse form of the Brunn-Minkowski inequality, with applications to the local theory of normed spaces]". Comptes Rendus de l'Académie des Sciences, Série I. 302 (1): 25–28. MR 0827101. Pisier, Gilles (1989). The volume of convex bodies and Banach space geometry. Cambridge Tracts in Mathematics. Vol. 94. Cambridge: Cambridge University Press. ISBN 0-521-36465-5. MR 1036275.

Worked examples

Example 1 — a first encounter with Milman's reverse Brunn–Minkowski inequality

Start with the simplest possible case. Write down what Milman's reverse Brunn–Minkowski inequality 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 Milman's reverse Brunn–Minkowski inequality 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 Milman's reverse Brunn–Minkowski inequality 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 Milman's reverse Brunn–Minkowski inequality

In research: Milman's reverse Brunn–Minkowski inequality 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 Milman's reverse Brunn–Minkowski inequality 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: Milman's reverse Brunn–Minkowski inequality is common in secondary-school and first-year university syllabi. It links to neighbouring topics Asymptotic geometric analysis, Euclidean geometry, Geometric inequalities, so understanding it makes those chapters shorter.
In everyday life: Look for Milman's reverse Brunn–Minkowski inequality 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 Milman's reverse Brunn–Minkowski inequality in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Milman's reverse Brunn–Minkowski inequality 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 Milman's reverse Brunn–Minkowski inequality out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Milman's reverse Brunn–Minkowski inequality in simple terms?

Why does Milman's reverse Brunn–Minkowski inequality 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 Milman's reverse Brunn–Minkowski inequality?

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 Milman's reverse Brunn–Minkowski inequality.