This is an open(?) problem. Let $K$ and $L$ be two compact, convex bodies in an $n-$dimensional Euclidean space. Let $P$ and $P'$ be any two $(n-1)$-dimensional orthogonal hyperplanes. Suppose $K$ and $L$ have the property that the projection of $K$ onto $P$ is similar to the projection of $L$ onto $P'$ AND the projection of $K$ onto $P'$ is similar to the projection of $L$ onto $P$. This property must hold for any two given orthogonal $(n-1)$-dimensional hyperplanes.
Claim: $K$ and $L$ are similar up to rotation, reflection, and translation.
I tried reducing the problem to the two dimensional case but projections are simply intervals and don't provide much information. In this case, if we compare the projection of a circle to that of a Meissner body, then I believe we get a contradiction. But does the same result hold in higher dimensions?