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 $\log h(\alpha)$ concave?

Is either 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 the answer to the above is negative. In case I am correct, I'd be happy to see an argument/example.

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.

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 $\log h(\alpha)$ concave?

Is either 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 the answer to the above is negative. In case I am correct, I'd be happy to see an argument/example.

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 $\log h(\alpha)$ concave?

Is either 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 the answer to the above is negative. In case I am correct, I'd be happy to see an argument/example.