If section is always contractible, is that convex?

Question

Consider compact set $\Omega\subset\mathbb{R}^d$, whose intersection with any $(d-1)$-dimensional subspace of $\mathbb{R}^d$ is contractible. Then, is such $\Omega$ convex?

Answer 1

No. Consider a large sphere, with a smaller sphere removed from the centre. The intersection with any plane is still connected (it will be a circle or an annulus). But this is clearly not a convex set. Similar higher-dimensional analogues can be found.