Do external electronic shells influence internal ones? There is a well-known electron shielding effect: negatively-charged internal electronic shell screens external shells from positive nuclei, thus increasing their radii. 

Thus, 2S orbital radius of Li atom (ground state) should be greater than radius of excited Li2+ cation with single 2S electron. 

But is there a reverse influence: from external electronic shells to internal ones? For example, are radii of 1S orbital in Li and Li+ equal, or not? 

I am sure there is not a big difference anyway, but this is theoretically interesting. Probably it can be proved by comparision of internal shell ionisation energies of Li and Li+ but I am not sure where to look for appropriate experimental data.

I have some intuitive considerations, while they did not lead me to certain answer: 


Concept of Faraday cage looks to be irrelevant, because electron
shells are not literally material conductive shells, but they are
filled space regions. 
Shells interpenetrate, thus external shell
can a bit force out internal shell from nucleus by increasing
electron density in space  adjacent to nucleus. 
Classically,
internal one of two concentric  elastic charged spheres should
shrink. If we imagine big spheres, their small areas should interact
repulsively as parallel plates. But considered spheres are actually
not "big".
Not only do "outer" electrons impart some shielding to "inner" ones, but an analytical technique is based on that fact.

In X-ray photoelectron spectroscopy, binding energies of electrons just below the outer shell are measured.  These binding energies are typically hundreds of electron volts, roughly 280-300 eV for carbon as a typical example.  An important feature is that this binding energy is measurably not fixed for a given element, rather is varies according to how the atom is chemically bound.  As an example, the 1s electrons in carbon lie at about 285 eV binding energy in hydrocarbons, but shifted by as much as 10 eV by bonding to fluorine (see here), which is more electronegative and draws electrons away from the carbon 2s and 2p  shells.  Similarly, the 1s electrons in oxygen range from about 529 to 534 eV in various oxides, with the higher values tending to occur in nonmetal oxides (see here).