What is a "geometric rectangle"?

Question

In the [Kushilevitz and Nisan 2006, P.10] they give an exercise which says as following:

Exercise 1.18: Let $X=Y=\{1,\ldots,n\}$. A geometric rectangle is a set of the form $\{(x,y)|x_{\min} \leq x \leq x_{\max}, y_{\min} \leq y \leq y_{\max}\}$, for some values $x_{\min},x_{\max},y_{\min}$, and $y_{\max}$ in $\{1,\ldots,n\}$. A comparison protocol is a one in which at each node $v$, if Alice needs to transmit a bit then this bit is the result of comparing her input $x$ with some value $\theta_v$ (that is, $a_v(x)$ is $0$ if $x < \theta_v$ and $1$ if $x \geq \theta_v$). Similarly, at each node $v$ where Bob speaks he sends the result of comparing his input $y$ with some value $\theta_v$. Prove that every comparison protocol for computing a function $f$ partitions the space $X\times Y$ into $f$-monochromatic geometric rectangles.

My simple question is "What is geometric rectangle" in order to understand the exercise?

Assume that we have 2 bits, then we have four possible ways to represent 2 bits on X and on Y, so total is $ 4 \times 4 = 16$ cells on the matrix. When I try to follow what the definition of "geometric rectangle", I found that all matrix have 1 entry; because it achieves the definition of a "geometric rectangle"; it didn't sounds "reasonable" because there is a comparison between the inputs, so there must be a {0} and {1} entry on the matrix.

I tried to look up for another definition of a "geometric rectangle", it turns out that "geometric rectangle" is "the Euclidean geometric rectangle", I don't understand what is the relation between "Euclidean geometric rectangle" and "combintorial rectangle" on communication complexity, does "geometric rectangle" refer to "Euclidean geometric rectangle" or there is misunderstanding here?

Any hence or reference would be very helpful, Thank You!

Answer 1

Geometric rectangle is not a standard term. That's why they define it in the question.

In communication complexity, Alice has an input from the set $X$, and Bob has an input from the set $Y$. Usually we assume that the sets $X,Y$ are unstructured. A combinatorial rectangle is a subset of $X \times Y$ of the form $A \times B$, where $A \subseteq X$ and $B \subseteq Y$.

In this question, the sets $X,Y$ are ordered, and the authors defined a geometric rectangle to be a subset of $X \times Y$ of the form $A \times B$ where $A$ is a subinterval of $X$ and $B$ is a subinterval of $Y$. That is, $A = \{ x \in X : x_1 \leq x \leq x_2 \}$ for some $x_1,x_2 \in X$, and $B = \{ y \in Y : y_1 \leq y \leq y_2 \}$ for some $y_1,y_2 \in Y$.