# Concavity of mixed volumes and mixed discriminants

For $n\times n$ symmetric matrices $A_1, \ldots, A_n$, the mixed discriminant $D(A_1, \ldots, A_n)$ can be defined as $1/n!$ times the coefficient of $t_1\ldots t_n$ in the homogeneous polynomial $\det(t_1A_1 + \ldots t_n A_n)$, i.e. $$D(A_1, \ldots, A_n) = \frac{\partial^n}{n!\partial t_1\ldots \partial t_n}\det(t_1A_1 + \ldots t_n A_n).$$

This is closely related to mixed volumes. If $K_1, \ldots, K_n$ are $n$ compact convex sets in $\mathbb{R}^n$, then the mixed volume $V(K_1, \ldots, K_n)$ is $1/n!$ times the coefficient of $t_1\ldots t_n$ in $\mathrm{vol}(t_1 K_1 + \ldots + t_n K_n)$, the latter being a homogeneous polynomial of degree $n$ by Minkowski's theorem. (Different authors sometimes omit the normalizing $1/n!$ term).

Both mixed discriminants and mixed volumes satisfy a generalized Brunn-Minkowski inequality. I.e. both functions below are concave on $\alpha \in [0,1]$: $$f(\alpha) = D(\alpha A + (1-\alpha)A', i; B_{i+1}, \ldots, B_n)^{1/i},\\ g(\alpha) = V(\alpha K + (1-\alpha)K', i; L_{i+1}, \ldots, L_n)^{1/i}.$$ Here I have used the notation $D(A, i; B_{i+1}, \ldots, B_n)$ for $D(A, \ldots, A, B_{i+1}, \ldots, B_n)$ with $i$ copies of $A$, and similarly for $V$. The concavity of these two functions is equivalent to the Aleksandrov-Fenchel inequalities. The point of this question is to understand how far this kind of concavity can be pushed, or is it the best possible.

Let me use the analogous notation $D(A, i; B, j; C_{i+j+1}, \ldots, C_n) = D(A, \ldots, A, B, \ldots, B, C_{i+j+1}, \ldots, C)$ with $i$ copies of $A$ and $j$ copies of $B$. My question(s) follow.

Is the function $h(\alpha) = D(\alpha A + (1-\alpha)A', i; \alpha B + (1-\alpha)B', j; C_{i+j+1}, \ldots, C_n)^{1/(i+j)}$ concave?

Is it concave if $A, A', B, B', C_{i+j+1}, \ldots, C_n$ are all positive semidefinite?

What about the analogous function with mixed volume in the place of mixed discriminant?

I suspect negative answers to the above. In case I am correct, I'd be happy to see an argument/example.

• The discussion on pg. 391 of ams.org/journals/bull/2002-39-03/S0273-0979-02-00941-2/… seems to contain useful pointers; the conjecture seems nice but probably it breaks. – Suvrit May 25 '15 at 14:27
• The closest paper I found relevant to your question is: arxiv.org/pdf/1412.8200.pdf --- does Corollary 4.1 in there help? – Suvrit May 25 '15 at 14:38
• @Suvrit Gardner's Bulletin paper is indeed a great resource. The other paper you link to talks about a different concavity question that goes back to Gromov. They fix $n$ bodies and look at mixed volume as a function on the discrete simplex $\{x \in \mathbb{Z}_+^n: \sum x_i = n\}$, and ask if the function is log concave. Aleksandrov-Fenchel says that it is log-concave on particular segments, and this turns out to be equivalent to generalized Brunn-Minkowski. However, I don't see a similar connection between the concavity of my function $h$ and concavity on the discrete simplex. – Sasho Nikolov May 25 '15 at 17:55
• For example, the simplest $i=j=1$ case of my question reduced to the inequality $D(A, B', C_3, \ldots, C_n)/2 + D(A',B, C_3, \ldots, C_n)/2 \ge \sqrt{D(A, B, C_3, \ldots, C_n)D(A',B',C_3, \ldots, C_n)}$. I dont see how to connect this to concavity of the mixed discriminant on the discrete simplex. – Sasho Nikolov May 25 '15 at 18:01
• Ok, that clarifies my doubts. Before plowing into this, however, I'd like to ask you do you have any numerical evidence that suggests that your conjecture may hold? Thanks! – Suvrit May 25 '15 at 19:25