A possible measure of asymmetry for a convex body $K \subset \mathbb{R}^n$ is the affine-invariant quantity $$ \alpha_n(K) := \frac{\textrm{vol}(K - K)}{2^n\textrm{vol}(K)} . $$ Indeed, the Brunn-Minkowski inequality states that $\alpha(K) \geq 1$ and that equality holds if and only if $K$ is centrally symmetric.

**Question.** *Can all the orthogonal projections $K|\zeta$ of an asymmetric convex body $K \subset \mathbb{R}^n$ onto different hyperplanes be "equally asymmetric" in the sense that
$\alpha_{n-1}(K|\zeta)$ is independent of the choice of hyperplane $\zeta$?*

I'm interested in bodies with $C^2$ boundary and strictly positive curvature so that, a priori, one can translate this problem into properties for Hessians of support functions and so forth, but the question sounds so elementary and basic that maybe the answer is simple or already known.

**Edit.** Exchanges with Dmitry Ryabogin led to the following reformulation of
the problem:

**Question.** *What are the convex bodies $K \subset \mathbb{R}^n$, $n > 2$, that are bodies of constant width and constant brightness relative to the gauge body $K - K$?*

The constant width part comes for free ($K$ is always of constant width relative to $K - K$), but in this form it is easy to see where we are:

- The answer seems unknown for general bodies even in dimension $n = 3$.
- For three-dimensional bodies whose boundary is $C^2$ and of strictly positive curvature, the answer is that $K$ must itself be centrally symmetric. This follows from a theorem of Chakerian (Theorem 6 in
*Sets of Constant Relative Width and Constant Relative Brightness*, Transactions of the AMS, 1967). - This is a weaker question than the classic problem of finding all bodies of constant relative width and brightness for a $0$-symmetric gauge body $B$, and the conjecture there is that the only such bodies are translates and dilates of $B$.