Asymmetry of projections

Question

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:

1. The answer seems unknown for general bodies even in dimension $n = 3$.
2. 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).
3. 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 $B$.

Answer 1

I am not sure if this is known. Two remarks.

1. Consider a slightly different problem, where your quantity is changed into the one that measures the surface area instead of the volume. Take a body of constant width in $R^3$, say, K. Then $K-K=B$ (Euclidean ball), i.e., the numerator for the new quantity is constant independently of the direction (all projections are again the bodies of the same constant width and $K|\xi-K|\xi=D$ with the radius of the disc $D$ independent of $\xi$ and equal to the radius of $B$). On the other hand, one can use the fact that $K$ is of constant width if and only if all projections have the same perimeter (i.e. the denominator for the new quantity is constant). It follows that there are bodies $K$ different from the Euclidean ball, that satisfy the property (but for the different measure of asymmetry).

2. It is not known if for dimensions $d\ge 4$, the only bodies which are of constant width and of constant brightness (both, the surface areas and the volumes of the projections are constant) are the Euclidean balls (it is known to be true in $R^3$). How to construct a body of constant brightness $K$ so that $vol_{d-1}(K|\xi-K\xi)$ is constant, say?

Please contact Daniel Hug, [email protected], he might know the answer.

Answer 2

This answer is a summary of discussions with Dmitry Ryabogin and Ralph Howard. The conclusion is that the original problem is equivalent to other long-standing open problems in convex geometry.

OP. Let $K \subset \mathbb{R}^n$ be a convex body such that the value of the measure of asymmetry $\alpha_{n-1}$ is the same for every one of its orthogonal projections onto hyperplanes. Does this imply that $K$ is centrally symmetric?

The next problem belongs to relative geometry, nowadays called Minkowski geometry (see Tony Thompson's book "Minkowski Geometry"). It is basically the geometry of submanifolds in finite-dimensional normed spaces.

Relative Nakajima Conjecture. Let $B \subset \mathbb{R}^n$ be a $0$-symmetric convex body. If another convex body $K$ has constant breadth and constant brightness with respect to $B$, then it is equal to $B$ up to translation and dilation.

The state of the art on this problem is Ralph Howard's solution when $B \subset \mathbb{R}^3$ is a regular gauge body (see his paper Convex bodies of constant width and constant brightness).

The last problem was formulated Rolf Schneider in 1981 and is Problem~3.5 p.~118 in Richard Gardner's book "Geometric Tomography".

Problem 3.5. Suppose that the Blaschke body $\nabla K$ of a convex body in $\mathbb{E}^n$ is homothetic to its central symmetral (its difference body) $\Delta K$. Is $K$ centrally symmetric?