Background

Take 2 convex sets in $\mathbb{R}^2$, or 3 convex sets in $\mathbb{R}^3$, or generally, $n$ convex sets in $\mathbb{R}^n$. "Mixed volume" assigns to such a family $A_1, \ldots, A_n$ a real number $V(A_1, \ldots, A_n)$, measured in $\mathrm{metres}^n$.

As I understand it, mixed volume is a kind of cousin of the determinant. I'll give the definition in a moment, but first here are some examples.

1. $V(A, \ldots, A) = \mathrm{Vol}(A)$, for any convex set $A$.

2. More generally, suppose that $A_1, \ldots, A_n$ are all scalings of a single convex set (so that $A = r_i B$ for some convex $B$ and $r_i \geq 0$). Then $V(A_1, \ldots, A_n)$ is the geometric mean of $\mathrm{Vol}(A_1), \ldots, \mathrm{Vol}(A_n)$.mathrm{Vol}(A_n)$. 3. The previous examples don't show how mixed volume typically depends on the interplay between the sets. So, taking$n = 2$, let$A_1$be an$a \times b$rectangle and$A_2$a$c \times d$rectangle in$\mathbb{R}^2$, with their edges parallel to the coordinate axes. Then $$V(A_1, A_2) = \frac{1}{2}(ad + bc).$$ (Compare and contrast the determinant formula$ad - bc$.) 4. More generally, take axis-parallel parallelepipeds $A_1, \ldots, A_n$ in$\mathbb{R}^n$. Write $m_{i1}, \ldots, m_{in}$ for the edge-lengths of$A_i$. Then $$V(A_1, \ldots, A_n) = \frac{1}{n!} \sum_{\sigma \in S_n} m_{1, \sigma(1)} \cdots m_{n, \sigma(n)}.$$ (Again, compare and contrast the determinant formula.) The definition of mixed volume depends on a theorem of Minkowski: for any compact convex sets $A_1, \ldots, A_m$ in$\mathbb{R}^n$, the function $$(\lambda_1, \ldots, \lambda_m) \mapsto \mathrm{Vol}(\lambda_1 A_1 + \cdots + \lambda_m A_m)$$ (where$\lambda_i \geq 0$and$+$means Minkowski sum) is a polynomial, homogeneous of degree$n$. For$m = n$, the mixed volume $V(A_1, \ldots, A_n)$ is defined as the coefficient of $\lambda_1 \lambda_2 \cdots \lambda_n$ in this polynomial, divided by$n!$. Why pick out this particular coefficient? Because it turns out to tell you everything, in the following sense: for any convex sets $A_1, \ldots, A_m$ in$\mathbb{R}^n$, $$\mathrm{Vol}(\lambda_1 A_1 + \cdots + \lambda_m A_m) = \sum_{i_1, \ldots, i_n = 1}^m V(A_{i_1}, \ldots, A_{i_n}) \lambda_{i_1} \cdots \lambda_{i_n}.$$ Properties of mixed volume Formally, let$\mathscr{K}_n$be the set of nonempty compact convex subsets of$\mathbb{R}^n$. Then mixed volume is a function $$V: (\mathscr{K}_n)^n \to [0, \infty),$$ and has the following properties: 1. Volume:$V(A, \ldots, A) = \mathrm{Vol}(A)$. (Here and below, the letters$A$,$A_i$etc. will be understood to range over$\mathscr{K}_n$, and$\lambda$,$\lambda_i$etc. will be nonnegative reals.) 2. Symmetry:$V$is symmetric in its arguments. 3. Multilinearity: $$V(\lambda A_1 + \lambda' A'_1, A_2, \ldots, A_n) = \lambda V(A_1, A_2, \ldots, A_n) + \lambda' V(A'_1, A_2, \ldots, A_n).$$ (These first three properties closely resemble a standard characterization of determinants.) 4. Continuity:$V$is continuous with respect to the Hausdorff metric on$\mathscr{K}_n$. 5. Invariance: $V(gA_1, \ldots, gA_n) = V(A_1, \ldots, A_n)$ for any isometry$g$of Euclidean space$\mathbb{R}^n$onto itself. 6. Multivaluation: $$V(A_1 \cup A'_1, A_2, \ldots, A_n) = V(A_1, A_2, \ldots) + V(A'_1, A_2, \ldots) - V(A_1 \cap A'_1, A_2, \ldots)$$ whenever $A_1, A'_1, A_1 \cup A'_1 \in \mathscr{K}_n$.mathscr{K}_n$.

7. Monotonicity: $V(A_1, A_2, \ldots, A_n) \leq V(B_1V(A'_1, A_2, \ldots, B_n)$ A_n)$ whenever $A_1 \subseteq B_1, \ldots, A_n \subseteq B_n$A'_1$.

There are other basic properties, but I'll stop there.

Questions

Is $V$ the unique function $(\mathscr{K}_n)^n \to [0, \infty)$ satisfying properties 1--7?

If so, does some subset of these properties suffice? In particular, do properties 1--3 suffice?

If not, is there a similar characterization involving different properties?

(Partway through writing this question, I found a recent paper of Vitali Milman and Rolf Schneider: Characterizing the mixed volume. I don't think it answers my question, though it does give me the impression that the answer might be unknown.)

1

# Do the elementary properties of mixed volume characterize it uniquely?

Background

Take 2 convex sets in $\mathbb{R}^2$, or 3 convex sets in $\mathbb{R}^3$, or generally, $n$ convex sets in $\mathbb{R}^n$. "Mixed volume" assigns to such a family $A_1, \ldots, A_n$ a real number $V(A_1, \ldots, A_n)$, measured in $\mathrm{metres}^n$.

As I understand it, mixed volume is a kind of cousin of the determinant. I'll give the definition in a moment, but first here are some examples.

1. $V(A, \ldots, A) = \mathrm{Vol}(A)$, for any convex set $A$.

2. More generally, suppose that $A_1, \ldots, A_n$ are all scalings of a single convex set (so that $A = r_i B$ for some convex $B$ and $r_i \geq 0$). Then $V(A_1, \ldots, A_n)$ is the geometric mean of $\mathrm{Vol}(A_1), \ldots, \mathrm{Vol}(A_n)$.

3. The previous examples don't show how mixed volume typically depends on the interplay between the sets. So, taking $n = 2$, let $A_1$ be an $a \times b$ rectangle and $A_2$ a $c \times d$ rectangle in $\mathbb{R}^2$, with their edges parallel to the coordinate axes. Then $$V(A_1, A_2) = \frac{1}{2}(ad + bc).$$ (Compare and contrast the determinant formula $ad - bc$.)

4. More generally, take axis-parallel parallelepipeds $A_1, \ldots, A_n$ in $\mathbb{R}^n$. Write $m_{i1}, \ldots, m_{in}$ for the edge-lengths of $A_i$. Then $$V(A_1, \ldots, A_n) = \frac{1}{n!} \sum_{\sigma \in S_n} m_{1, \sigma(1)} \cdots m_{n, \sigma(n)}.$$ (Again, compare and contrast the determinant formula.)

The definition of mixed volume depends on a theorem of Minkowski: for any compact convex sets $A_1, \ldots, A_m$ in $\mathbb{R}^n$, the function $$(\lambda_1, \ldots, \lambda_m) \mapsto \mathrm{Vol}(\lambda_1 A_1 + \cdots + \lambda_m A_m)$$ (where $\lambda_i \geq 0$ and $+$ means Minkowski sum) is a polynomial, homogeneous of degree $n$. For $m = n$, the mixed volume $V(A_1, \ldots, A_n)$ is defined as the coefficient of $\lambda_1 \lambda_2 \cdots \lambda_n$ in this polynomial, divided by $n!$.

Why pick out this particular coefficient? Because it turns out to tell you everything, in the following sense: for any convex sets $A_1, \ldots, A_m$ in $\mathbb{R}^n$, $$\mathrm{Vol}(\lambda_1 A_1 + \cdots + \lambda_m A_m) = \sum_{i_1, \ldots, i_n = 1}^m V(A_{i_1}, \ldots, A_{i_n}) \lambda_{i_1} \cdots \lambda_{i_n}.$$

Properties of mixed volume

Formally, let $\mathscr{K}_n$ be the set of nonempty compact convex subsets of $\mathbb{R}^n$. Then mixed volume is a function $$V: (\mathscr{K}_n)^n \to [0, \infty),$$ and has the following properties:

1. Volume: $V(A, \ldots, A) = \mathrm{Vol}(A)$. (Here and below, the letters $A$, $A_i$ etc. will be understood to range over $\mathscr{K}_n$, and $\lambda$, $\lambda_i$ etc. will be nonnegative reals.)

2. Symmetry: $V$ is symmetric in its arguments.

3. Multilinearity: $$V(\lambda A_1 + \lambda' A'_1, A_2, \ldots, A_n) = \lambda V(A_1, A_2, \ldots, A_n) + \lambda' V(A'_1, A_2, \ldots, A_n).$$ (These first three properties closely resemble a standard characterization of determinants.)

4. Continuity: $V$ is continuous with respect to the Hausdorff metric on $\mathscr{K}_n$.

5. Invariance: $V(gA_1, \ldots, gA_n) = V(A_1, \ldots, A_n)$ for any isometry $g$ of Euclidean space $\mathbb{R}^n$ onto itself.

6. Multivaluation: $$V(A_1 \cup A'_1, A_2, \ldots, A_n) = V(A_1, A_2, \ldots) + V(A'_1, A_2, \ldots) - V(A_1 \cap A'_1, A_2, \ldots)$$ whenever $A_1, A'_1, A_1 \cup A'_1 \in \mathscr{K}_n$.

7. Monotonicity: $V(A_1, \ldots, A_n) \leq V(B_1, \ldots, B_n)$ whenever $A_1 \subseteq B_1, \ldots, A_n \subseteq B_n$.

There are other basic properties, but I'll stop there.

Questions

Is $V$ the unique function $(\mathscr{K}_n)^n \to [0, \infty)$ satisfying properties 1--7?

If so, does some subset of these properties suffice? In particular, do properties 1--3 suffice?

If not, is there a similar characterization involving different properties?

(Partway through writing this question, I found a recent paper of Vitali Milman and Rolf Schneider: Characterizing the mixed volume. I don't think it answers my question, though it does give me the impression that the answer might be unknown.)