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Minkowski's first inequality for convex bodies

Let K and L be two n-dimensional convex bodiesinn-dimensional Euclidean space Rⁿ. Define a quantity V₁(K, L) by

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

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

${\displaystyle V_{1}(K,L)\geq V($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[edit]

• V₁ is just one example of a class of quantities known as mixed volumes.
• IfL 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[edit]

The Brunn–Minkowski inequality[edit]

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[edit]

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

${\displaystyle \left({\frac {V($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[edit]

• 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.

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