  Let $P\subset\mathbb{R}^n$ $(n\geq 3)$ be a convex polytope containing the
origin in its interior. Let $\mbox{vol}_{n-2} \big( \mbox{relbd} ( P\cap\lbrace
t\xi + \xi^\perp \rbrace ) \big)$ denote the $(n-2)$-dimensional volume of the
relative boundary of $P\cap\lbrace t\xi + \xi^\perp \rbrace$ for
$t\in\mathbb{R}$, $\xi\in S^{n-1}$. We prove the following: if \begin{align*}
\mbox{vol}_{n-2} \Big( \mbox{relbd} \big( P\cap\xi^\perp \big) \Big)
  = \max_{t\in\mathbb{R}} \mbox{vol}_{n-2} \Big( \mbox{relbd} \big(
P\cap\lbrace t\xi + \xi^\perp \rbrace \big) \Big)
  \ \ \forall \ \ \xi\in S^{n-1}, \end{align*} then $P$ is origin-symmetric,
i.e. $P = -P$. Our result gives a partial affirmative answer to a conjecture by
Makai, Martini, and \'Odor. We also characterize the origin-symmetry of $C^1$
convex bodies in terms of the dual quermassintegrals of their sections; this
can be seen as a dual version of the conjecture of Makai et al.
