Generalizing quadratic $\mathbb{R}$-Algebraic sets in $\mathbb{CP}^{n}$

Motivated by our study of the complex Banach conjecture, we characterize a complex ellipsoids $\mathcal E$ as compact subsets of $\mathbb C^n$, with the property that every complex line intersect $\mathcal E$ either in a single point or in the complex affine image of the unit disk. This characterization leads to the main interest of this paper. We study the topological behavior of compact subsets of $\mathbb{CP}^n$ with the property that any complex line that intersects them does either at a single point, at the boundary of a complex disk, or along the entire line. In particular, we are interested in quadratic $\R$-algebraic subsets of $\mathbb{CP}^n$.