We investigate the geometrical structure of the emitting part of circu
mstellar envelopes around Be stars from an empirical point of view. We
use new high-resolution, high-S/N spectroscopic data of the Fe II lam
bda 5317 and some other faint Fe II emission lines in 27 Be stars show
ing symmetrical emission lines (class 1). We find a clear correlation
between its total width (measuring the maximum velocities of circumste
llar matter) and the stellar rotational velocity. This correlation mea
ns that a typical Be envelope (or, more precisely, that part of it whi
ch is visible in optical emission lines) is an axisymmetric, rotationa
lly supported disk. For empirical investigation of the vertical struct
ure, we use the occurrence of shell lines. We define, as shell criteri
on based on Fe II lines, a Be shell star as one with Fe II central int
ensity F-cd/F(Fe II) < 1. Using this for calibrating an appropriate p
arameter for the much more frequently observed H alpha line, we find t
hat shell stars are those with F-p/F-cd(H alpha) greater than or equal
to 1.5 where F-p is the mean peak intensity at H alpha. In a sample o
f 114 programme stars, we find a shell star fraction of 22.8%. This nu
mber is readily transformed into a half opening angle of Be star disks
, phi = 13 degrees. We furthermore show that Be disks must be thin at
the inner edge, and may become fairly thick at the outer rim. This, to
gether with the small value of phi, is evidence for a conical or conca
ve shape, the latter typical of a hydrostatically balanced disk. Final
ly we provide evidence that the famous ''shell-Be'' phase transitions
can naturally occur in such disks as a geometrical effect if they are
seen under inclination i approximate to 70 degrees and if their outer
radius is variable with time.