This is a more interesting follow-up to the question Are closed simple curves with this property necessarily circles?
Let $\gamma:[0,1]\to \mathbb R^2 $ be a closed simple $C^1$ convex curve and $\Gamma$ be the region enclosed by $\gamma$. Let $O$ be the center of mass of $\Gamma$.
Suppose that any two perpendicular lines that go through $O$ split $\gamma$ into four regions with equal areas.
Is $\gamma$ a circle ?
Again, I'd say the answer is yes, but I'm looking for a rigorous proof.
As commenters pointed out, any smooth curve with 4-fold symmetry is a counterexample, such as $\gamma = \{(x,y): x^4+y^4 = 1\}$.
Indeed, 4-fold symmetry means there is a point $C$ such that rotation by $90$ degrees about $C$ maps $\Gamma$ to itself. Consequently, such rotation maps the center of mass $O$ to itself, which implies $O=C$.
Since rotation by $90$ degrees about $O$ maps the union of lines onto itself, the four regions into which they divide $\Gamma$ are mapped onto each other. Thus, they have equal area.