In 1970 or so, Klee asked if a convex body in $\mathbb R^n$ ($n\ge 3$) whose maximal sections by hyperplanes in all directions have the same volume must be a ball. The counterexample in $\mathbb R^4$ is trivial and can be described as follows:
Let $f:[-1,1]\to\mathbb R$ be continuous, strictly concave and satisfy $f(-1)=f(1)=0$. For every such function, let $Q_f$ be the body of revolution given by $y^2+z^2+t^2\le f(x)^2$. Then $Q_f$ and $Q_g$ have the same maximal sections in every direction if (and, actually, only if) $f$ and $g$ are equimeasurable, i.e., $|{f>t}|=|{g>t}|$ for all $t>0$. (Of course, there are plenty of concave functions equimeasurable with $\sqrt{1-x^2}$).
With some extra work, one can construct something like this in $\mathbb R^n$ when $n$ is even though I do not know any similarly nice geometric description of such bodies for $n\ge 6$.
What I (and my co-authors) are currently stuck with is the case of odd $n$ (say, the usual space $n=3$). In view of such simple example in $\mathbb R^4$, I suspect that we are just having a mental block. Can anybody help us out?
