In The Cross-Section Body, Plane Sections of Convex Bodies and Approximation of Convex Bodies, I (Makai and Martini, 1996), the authors define for a convex body $K$ its cross-section body $CK$ and its projection body $\Pi K$.
Both bodies, as I understand it, are defined in terms of their support function $h(u)$ specifying the extent of the body from the origin in direction $u\in S^{n-1}$.
- For $CK$ we have $h(u)=\sup_{\lambda\in\mathbb R}V_{d-1}(K\cap (u^\perp + \lambda u))$, where $u^\perp$ is the orthogonal plane to $u$ and $V_{d-1}$ is the $(d-1)$-dimensional volume.
- For $\Pi K$ we have $h(u)$ equal to the $(d-1)$-volume of the orthogonal projection of $K$ onto $u^\perp$.
The authors state that $CK\subset \Pi K$, which makes sense to me; the maximal cross-section in a direction will obviously have at most as much area as the projection from that direction.
They also state, however, that $CK=\Pi K$ for $K$ an ellipsoid when $n\ge 3$, which confuses me. This would seem to me to imply that for each $u\in S^{d-1}$, we have that the maximal-area cross section orthogonal to $u$ occupies as much area (or $(d-1)$-volume) as the projection onto $u$.
But this seems clearly false! Consider a highly prolate spheroid, such as $x^2+y^2+0.1z^2 = 1$, and take e.g. $u=(1,1,1)/\sqrt{3}$. Then it seems obvious to me that the maximal cross-section cannot attain as much area as the projection - just looking at things visually, as we translate the intersecting plane we must choose whether to include either end of the ellipse along its longest axis, and will inevitably miss out on parts of the projection covered only by the other end of the ellipse.
(In fact, the same effect shows up even in 2D ellipses, for any direction besides the axes of the ellipse.)
For this claim, they cite Martini 1989, but looking at that paper I'm still confused: they refer for this claim to the statement that
For $d \ge 3$ Blaschke proved that a convex body $K$ is a $d$-ellipsoid if, and only if, for each direction $s \in S^{d-1}$ its shadow-boundary $S \cap \partial K$ is contained in a hyperplane.
But this doesn't mean that the shadow-boundary is contained in a hyperplane orthogonal to $s$, just that it's in some hyperplane; I don't see why the claim given suffices to show their result.
What am I misunderstanding here? The cited 1989 paper of Martini also goes to some work to show results that I thought would have been trivial (in particular I don't see why their Theorem with equation (2) isn't an immediate consequence of the inequality $\underline V_{d-1}(K,u)\le \overline{V}_{d-1}(K,u)$), so I suspect I have some terminological confusion here or am misunderstanding some notation. Any clarification would be appreciated!