Area in real K3-surfaces

For a real K3-surface $X$, one can introduce areas of connected components of the real point set $R X$ of $X$ using a holomorphic symplectic form of $X$. These areas are defined up to simultaneous multiplication by a real number, so the areas of different components can be compared. For any real K3-surface admitting a suitable polarization of degree $2g - 2$ (where $g$ is a positive integer) and such that $R X$ has one non-spherical component and at least $g$ spherical components, we prove that the area of the non-spherical component is greater than the area of every spherical component. We also prove that, for any real K3-surface with a real component of genus at least two, the area of this component is greater than the area of every spherical real component.