This is somewhat orthogonal to your question, but the following characterization of compact convex sets was proved by Atiyah in Convexity and commuting Hamiltonians. Bull. London Math. Soc. 14 (1982), no. 1, 1–15 using Morse theory:

A compact set $K\subset\mathbb{R}^n$ is convex $\iff$ its image $p(K)$ under any linear projection $p:\mathbb{R}^n \to\mathbb{R}^k$ is connected.

Edit Corrected.

Here is an elementary statement with a similar flavor (this does not answer the question).

Let $M$ be a compact manifold and $\mathcal{F}$ a family of functions $f: M\to \mathbb{R}^{n_f}$ with the following properties:

a. $\mathcal{F}$ is closed under linear projections

b. The pre-image $f^{-1}(c)\subset M$ under $f\in\mathcal{F}$ of any point $c\in \mathbb{R}^{n_f}$ is either empty or connected.

Then the image $f(M)$ of $M$ under any $f\in\mathcal{F}$ is convex.

Using Morse theory, Atiyah proved that these hypotheses hold for the moment maps of Hamiltonian torus actions on a compact symplectic manifold $M$¹ and concluded that the image $\mu(M)$ is a convex polytope whose vertices are the images of the fixed points.

Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), no. 1, 1–15

Footnote

¹ More precisely, families of commuting Hamiltonians which generate a compact subgroup in $Diff(M).$