Let $A=(A_{ij})_{1\le i,j\le n}$ be a square matrix with nonnegative real entries. Recall that $A$ is called a substochastic matrix if $$ \forall i,\ \ \sum_j A_{ij}\le 1\ . $$ In the course of my recent work on multidimensional resultants, I was led to the following determinantal inequality $$ \lvert\det(A-B)\rvert\le 1\ , $$ which must hold for every substochastic matrices $A$ and $B$.

My first question is: did this inequality appear previously in the literature?

Note that as Darij said in a comment, the inequality can also be stated as, for every $n\times n$ matrix $M=(M_{ij})_{1\le i,j\le n}$ with real entries $$ \lvert\det(M)\rvert\le \prod_{i=1}^{n}\lVert M_{i\ast}\rVert $$ where $M_{i\ast}$ is the $i$-th row of $M$, and the norm $\lVert x\rVert$ of a vector $x=(x_1,\dotsc,x_n)$ is defined by $$ \lVert x\rVert=\max_{J\subset[n]}\left|\sum_{j\in J}x_j\right|\ . $$ This formulation is reminiscent of the notion of total variation distance between discrete probability measures.

My second question is: can one classify the equality cases?

Let $A=(A_{ij})_{1\le i,j\le n}$ be a square matrix with nonnegative real entries. Recall that $A$ is called a substochastic matrix if $$ \forall i,\ \ \sum_j A_{ij}\le 1\ . $$ In the course of my recent work on multidimensional resultants, I was led to the following determinantal inequality $$ \lvert\det(A-B)\rvert\le 1\ , $$ which must hold for every substochastic matrices $A$ and $B$.

My first question is: did this inequality appear previously in the literature?

Note that as Darij said in a comment, the inequality can also be stated as, for every $n\times n$ matrix $M=(M_{ij})_{1\le i,j\le n}$ with real entries $$ \lvert\det(M)\rvert\le \prod_{i=1}^{n}\lVert M_{i\ast}\rVert $$ where $M_{i\ast}$ is the $i$-th row of $M$, and the norm $\lVert x\rVert$ of a vector $x=(x_1,\dotsc,x_n)$ is defined by $$ \lVert x\rVert=\max_{J\subset[n]}\left|\sum_{j\in J}x_j\right|\ . $$ This formulation is reminiscent of the notion of total variation distance between discrete probability measures.

My second question is: can one classify the equality cases, or even better, can one classify extremal pairs of transversals defined in section 4 of the preprint mentioned above?

Let $A=(A_{ij})_{1\le i,j\le n}$ be a square matrix with nonnegative real entries. Recall that $A$ is called a substochastic matrix if $$ \forall i,\ \ \sum_j A_{ij}\le 1\ . $$ In the course of my recent work on multidimensional resultants, I was lead to the following determinantal inequality $$ |{\rm det}(A-B)|\le 1\ , $$ which must hold for every substochastic matrices $A$ and $B$.

My first question is: did this inequality appear previously in the literature?

Note that as Darij said in a comment, the inequality can also be stated as, for every $n\times n$ matrix $M=(M_{ij})_{1\le i,j\le n}$ with real entries $$ |{\rm det}(M)|\le \prod_{i=1}^{n}\|M_{i\ast}\| $$ where $M_{i\ast}$ is the $i$-th row of $M$, and the norm $\|x\|$ of a vector $x=(x_1,\ldots,x_n)$ is defined by $$ \|x\|=\max_{J\subset[n]}\left|\sum_{j\in J}x_j\right|\ . $$ This formulation is reminiscent of the notion of total variation distance between discrete probability measures.

My second question is: can one classify the equality cases.

Let $A=(A_{ij})_{1\le i,j\le n}$ be a square matrix with nonnegative real entries. Recall that $A$ is called a substochastic matrix if $$ \forall i,\ \ \sum_j A_{ij}\le 1\ . $$ In the course of my recent work on multidimensional resultants, I was led to the following determinantal inequality $$ \lvert\det(A-B)\rvert\le 1\ , $$ which must hold for every substochastic matrices $A$ and $B$.

My first question is: did this inequality appear previously in the literature?

Note that as Darij said in a comment, the inequality can also be stated as, for every $n\times n$ matrix $M=(M_{ij})_{1\le i,j\le n}$ with real entries $$ \lvert\det(M)\rvert\le \prod_{i=1}^{n}\lVert M_{i\ast}\rVert $$ where $M_{i\ast}$ is the $i$-th row of $M$, and the norm $\lVert x\rVert$ of a vector $x=(x_1,\dotsc,x_n)$ is defined by $$ \lVert x\rVert=\max_{J\subset[n]}\left|\sum_{j\in J}x_j\right|\ . $$ This formulation is reminiscent of the notion of total variation distance between discrete probability measures.

My second question is: can one classify the equality cases?

Let $A=(A_{ij})_{1\le i,j\le n}$ be a square matrix with nonnegative real entries. Recall that $A$ is called a substochastic matrix if $$ \forall i,\ \ \sum_j A_{ij}\le 1\ . $$ In the course of my recent work on resultants, I was lead to the following determinantal inequality $$ |{\rm det}(A-B)|\le 1\ , $$ which must hold for every substochastic matrices $A$ and $B$.

My first question is: did this inequality appear previously in the literature?

Note that as Darij said in a comment, the inequality can also be stated as, for every $n\times n$ matrix $M=(M_{ij})_{1\le i,j\le n}$ with real entries $$ |{\rm det}(M)|\le \prod_{i=1}^{n}\|M_{i\ast}\| $$ where $M_{i\ast}$ is the $i$-th row of $M$, and the norm $\|x\|$ of a vector $x=(x_1,\ldots,x_n)$ is defined by $$ \|x\|=\max_{J\subset[n]}\left|\sum_{j\in J}x_j\right|\ . $$ This formulation is reminiscent of the notion of total variation distance between discrete probability measures.

My second question is: can one classify the equality cases.

Let $A=(A_{ij})_{1\le i,j\le n}$ be a square matrix with nonnegative real entries. Recall that $A$ is called a substochastic matrix if $$ \forall i,\ \ \sum_j A_{ij}\le 1\ . $$ In the course of my recent work on multidimensional resultants, I was lead to the following determinantal inequality $$ |{\rm det}(A-B)|\le 1\ , $$ which must hold for every substochastic matrices $A$ and $B$.

My first question is: did this inequality appear previously in the literature?

Note that as Darij said in a comment, the inequality can also be stated as, for every $n\times n$ matrix $M=(M_{ij})_{1\le i,j\le n}$ with real entries $$ |{\rm det}(M)|\le \prod_{i=1}^{n}\|M_{i\ast}\| $$ where $M_{i\ast}$ is the $i$-th row of $M$, and the norm $\|x\|$ of a vector $x=(x_1,\ldots,x_n)$ is defined by $$ \|x\|=\max_{J\subset[n]}\left|\sum_{j\in J}x_j\right|\ . $$ This formulation is reminiscent of the notion of total variation distance between discrete probability measures.

My second question is: can one classify the equality cases.