If area→zero does diameter of region→zero?

Question

I know that if region varies and diameter→0 then area→0 converse is not true how?

Answer 1

As you didn't specify a concrete context, I will use the following definition of diameter $$\mathrm{diam}(X):=\sup\{d(x,\tilde{x})\,|\,x,\tilde{x}\in X\}$$ together with the metric $d$ that will be the Euclidean distance of $\mathbb{R}^2$, i.e., the one given by $d(x,y):=\sqrt{(x_1-y_1)^2+(x_2-y_2)^2}$. Say if it differs.

In this context, consider $$X_t:=[0,\sqrt{t}]\times \left[0,\frac{1}{t}\right]$$ for $t>0$. Then as $t\to \infty$, you can see that $$\mathrm{diam}(X_t)=\sqrt{t}\to\infty\text{,}$$ as $(0,0),(0,\sqrt{t})\in X_t$, and that $$\mathrm{area}(X_t)=\frac{1}{\sqrt{t}}\to 0\text{.}$$ Hence as $t\to \infty$, not only does the area goes to zero while the diameter doesn't, but the area goes to zero as the diameter goes to infinity. Further, observe that the region $X_t$ "varies continuously" with respect $t$ (under a reasonable definition I will not enter in).

In conclusion, the trick is that a region can be very wide in one direction and very narrow in the other. This permits one to construct object that are narrow enough in the other direction, so that the area becomes small.