Question: Let $A$ be a compact set in $\mathbb R^n$. Is it always true that $\text{mean-width}(A) \ge C \cdot \text{diam}(A)$ for some constant $C$ depending only on the dimension? If not, is it true assuming $A$ is convex?
Background / definitions:
Given a compact set $A\subset\mathbb R^n$ and a direction $v\in\mathcal{S}^{n-1}$, the width $b(A, v)$ is the distance between two supporting hyperplanes of $A$ in the directions $v$ and $-v$, as illustrated in the following figure borrowed from M. Moszynska's Selected topics in convex geometry:
One can define the minimal width, diameter and mean width of $A$ as the minimum value of $b(A, v)$, the maximum value of $b(A, v)$ and the integral mean over $v\in\mathcal S^{n-1}$ of $b(A,v)$, respectively.
The following is obvious:
$$\text{min-width}(A) \le \text{mean-width}(A) \le \text{diam}(A)$$
It is also easy to see that the min-width and the diameter are "independent": They can be equal (i.e. in a ball), or they can be very different (i.e. a long thin tube).
However for the mean-width I don't understand the situation. Can it be that the mean width is arbitrarily small with a fixed diameter?
Let $L$ be any line segment; then $$ \frac{w(A)}{\text{diam}(A)} \ge \frac{w(L)}{\text{diam}(L)} $$ (where $w(\cdot)$ denotes mean width). Note that $w/\text{diam}$ is invariant under scaling, translation, and rotation, so the value of the RHS doesn't depend on which line segment you choose. In particular, the RHS depends only on the dimension, as desired (and in fact it turns out to be roughly $1/\sqrt{2\pi n}$). To prove the inequality, take $L$ to be a line segment joining two points in $A$ that realize the diameter of $A$, and then note $A\supseteq L$ (if $A$ is convex, which we may assume wlog since replacing $A$ with its convex hull doesn't change its widths, or its diameter, or its compactness).
(This argument, slightly generalized, appears in A. D. Aleksandrov, "On Mean Values of Support Functions", Soviet Math. Dokl. 8 (1967), 149–153, but I wouldn't be at all surprised to learn that it was known decades earlier.)