This question comes from my real analysis class.
For any $v=(\cos\theta,\sin\theta) ,\theta\in[0,2\pi)$, let $\pi_{v}(x)=\langle v, x\rangle . $ Let $A, B$ be two bounded open convex sets in $\mathbb{R}^{2}$ and $\mu$ be the Lebesgue measure on $\mathbb{R}^{1}$. If for all $v,\mu\left(\pi_{v}(A)\right)=\mu\left(\pi_{v}(B)\right)$, can we conclude that $A$ differs from $B$ by a translation, or rotation, or a reflection?
My idea is that since bounded open convex set is connected and $\pi_{v}(x)$ is continuous, we can only concern about the value on $\partial A$ and $\partial B$. Then I try to use the arc-length parametrization of the boundary curve to establish some equations, but I don't know how to continue. Any help is greatly appreciated!
No. For instance, using lengths of projections you cannot tell apart an open unit disk and (the interior of) a Reuleaux triangle of the constant width 2.