  The central surface brightness ($\mu_V(0)$) of the dwarf spheroidal
satellites of our own Galaxy is found to correlate with their distance from the
Galactic Center ($R_{GC}$). This observational evidence suggests that
environmental effects could strongly influence their structure. This suggestion
is further supported by a clear-cut bivariate correlation linking $\mu_V(0)$ to
a linear combination of $M_V$, the absolute integrated magnitude, and $R_{GC}$,
which seems to be shared also by the M31 spheroidals. Possible implications for
the Fundamental Plane of elliptical galaxies and Dark Matter content of dwarf
spheroidals are also briefly discussed.
