Mixed volume

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In mathematics, more specifically, in convex geometry, the mixed volume is a way to associate a non-negative number to an n-tuple of convex bodies in the n-dimensional space. This number depends on the size of the bodies and their relative positions.[1]

Definition[edit]

Let K1K2, ..., Kr be convex bodies in Rn, and consider the function

 f(\lambda_1, \ldots, \lambda_r) 
= \mathrm{Vol}_n (\lambda_1 K_1 + \cdots + \lambda_r K_r), \qquad \lambda_i \geq 0,

where Voln stands for the n-dimensional volume and its argument is the Minkowski sum of the scaled convex bodies Ki. One can show that f is a homogeneous polynomial of degree n, therefore it can be written as

 f(\lambda_1, \ldots, \lambda_r)
 = \sum_{j_1, \ldots, j_n = 1}^r V(K_{j_1}, \ldots, K_{j_n}) 
   \lambda_{j_1} \cdots \lambda_{j_n},

where the functions V are symmetric. Then V(T1, ..., Tn) is called the mixed volume of T1T2, ..., Tn.

Equivalently,


V(T_1, \ldots, T_n) 
= \left. \frac{\partial^n}{\partial \lambda_1 \cdots \partial \lambda_n}\right|_{\lambda_1 = \cdots = \lambda_n = +0} 
  \mathrm{Vol}_n(\lambda_1 T_1 + \cdots + \lambda_n T_n).

Properties[edit]

  • The mixed volume is uniquely determined by the following three properties:
  1. V(T, ...., T) = Voln(T);
  2. V is symmetric in its arguments;
  3. V is multilinear: V(a T + b ST2, ..., Tn) =a V(TT2, ..., Tn) + b V(ST2, ..., Tn) for a,b ≥ 0.
  • The mixed volume is non-negative, and increasing in each variable.
 V(T_1, T_2, T_3, \ldots, T_n) \geq \sqrt{V(T_1, T_1, T_3, \ldots, T_n) V(T_2,T_2, T_3,\ldots,T_n)}.
Numerous geometric inequalities, such as the Brunn–Minkowski inequality for convex bodies and Minkowski's first inequality, are special cases of the Alexandrov–Fenchel inequality.

Quermassintegrals[edit]

Let K ⊂ Rn be a convex body, and let B ⊂ Rn be the Euclidean ball. The mixed volume

 W_j(K) = V(\overset{n-j \text{ times}}{\overbrace{K,K, \ldots,K}}, \overset{j \text{ times}}{\overbrace{B,B,\ldots,B}})

is called the j-th quermassintegral of K.[2]

The definition of mixed volume yields the Steiner formula (named after Jakob Steiner):

 \mathrm{Vol}_n(K + tB)
 = \sum_{j=0}^n \binom{n}{j} W_j(K) t^j.

Intrinsic volumes[edit]

The j-th intrinsic volume of K is defined by

 V_j(K) = \binom{n}{j} \frac{W_{n-j}(K)}{\kappa_{n-j}},

where κnj is the volume of the (n − j)-dimensional ball.

Hadwiger's characterization theorem[edit]

Main article: Hadwiger's theorem

Hadwiger's theorem asserts that every valuation on convex bodies in Rn that is continuous and invariant under rigid motions of Rn is a linear combination of the quermassintegrals (or, equivalently, of the intrinsic volumes).[3]

Notes[edit]

  1. ^ Burago, Yu.D. (2001), "Mixed volume theory", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 
  2. ^ McMullen, P. (1991). "Inequalities between intrinsic volumes". Monatsh. Math. 111 (1): 47–53. doi:10.1007/bf01299276. MR 1089383. 
  3. ^ Klain, D.A. (1995). "A short proof of Hadwiger's characterization theorem". Mathematika 42 (2): 329–339. doi:10.1112/s0025579300014625. MR 1376731.