Mixed volume

 ( Convex Geometry ) 

In mathematics, more specifically, in convex geometry, the mixed volume is a way to associate a non-negative number to a tuple of convex body in $\backslash mathbb\{R\}^n$. This number depends on the size and shape of the bodies, and their relative orientation to each other.

---

Definition Let $K\_1,\; K\_2,\; \backslash dots,\; K\_r$ be convex bodies in $\backslash mathbb\{R\}^n$ and consider the function

$f(\backslash lambda\_1,\; \backslash ldots,\; \backslash lambda\_r)$

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

where $\backslash text\{Vol\}\_n$ stands for the $n$-dimensional volume, and its argument is the Minkowski sum of the scaled convex bodies $K\_i$. One can show that $f$ is a homogeneous polynomial of degree $n$, so can be written as

$f(\backslash lambda\_1,\; \backslash ldots,\; \backslash lambda\_r)$

= \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. For a particular index function $j\; \backslash in\; \backslash \{1,\backslash ldots,r\backslash \}^n$, the coefficient $V(K\_\{j\_1\},\; \backslash dots,\; K\_\{j\_n\})$ is called the mixed volume of $K\_\{j\_1\},\; \backslash dots,\; K\_\{j\_n\}$.

---

Properties

• The mixed volume is uniquely determined by the following three properties:
• $$

V(K, \dots, K) =\text{Vol}_n (K);

1. $V$ is symmetric in its arguments;
2. $V$ is multilinear: $$

V(\lambda K + \lambda' K', K_2, \dots, K_n) = \lambda V(K, K_2, \dots, K_n) + \lambda' V(K', K_2, \dots, K_n) for $\backslash lambda,\backslash lambda\text{'}\; \backslash geq\; 0$.

• The mixed volume is non-negative and monotonically increasing in each variable: $$

V(K_1, K_2, \ldots, K_n) \leq V(K_1', K_2, \ldots, K_n) for $K\_1\; \backslash subseteq\; K\_1\text{'}$.

• The Alexandrov–Fenchel inequality, discovered by Aleksandr Danilovich Aleksandrov and Werner Fenchel:

:$V(K\_1,\; K\_2,\; K\_3,\; \backslash ldots,\; K\_n)\; \backslash geq\; \backslash sqrt\{V(K\_1,\; K\_1,\; K\_3,\; \backslash ldots,\; K\_n)\; V(K\_2,K\_2,\; K\_3,\backslash ldots,K\_n)\}.$

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 Let $K\; \backslash subset\; \backslash mathbb\{R\}^n$ be a convex body and let $B\; =\; B\_n\; \backslash subset\; \backslash mathbb\{R\}^n$ be the Euclidean ball of unit radius. The mixed volume

$W\_j(K)\; =\; V(\backslash overset\{n-j\; \backslash text\{\; times\}\}\{\backslash overbrace\{K,K,\; \backslash ldots,K\}\},\; \backslash overset\{j\; \backslash text\{\; times\}\}\{\backslash overbrace\{B,B,\backslash ldots,B\}\})$

is called the j-th quermassintegral of $K$.

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

$\backslash mathrm\{Vol\}\_n(K\; +\; tB)$

= \sum_{j=0}^n \binom{n}{j} W_j(K) t^j.
     

---

Intrinsic volumes The j-th intrinsic volume of $K$ is a different normalization of the quermassintegral, defined by

$V\_j(K)\; =\; \backslash binom\{n\}\{j\}\; \backslash frac\{W\_\{n-j\}(K)\}\{\backslash kappa\_\{n-j\}\},$ or in other words $\backslash mathrm\{Vol\}\_n(K\; +\; tB)\; =\; \backslash sum\_\{j=0\}^n\; V\_j(K)\backslash ,\; \backslash mathrm\{Vol\}\_\{n-j\}(tB\_\{n-j\})\; =\; \backslash sum\_\{j=0\}^n\; V\_j(K)\backslash ,\backslash kappa\_\{n-j\}t^\{n-j\}.$

where $\backslash kappa\_\{n-j\}\; =\; \backslash text\{Vol\}\_\{n-j\}\; (B\_\{n-j\})$ is the volume of the $(n-j)$-dimensional unit ball.

---

Hadwiger's characterization theorem Hadwiger's theorem asserts that every valuation on convex bodies in $\backslash mathbb\{R\}^n$ that is continuous and invariant under rigid motions of $\backslash mathbb\{R\}^n$ is a linear combination of the quermassintegrals (or, equivalently, of the intrinsic volumes).

---

Notes

---

External links

Post Comment