'sudo gedit' stopped working

[Patrick Wiseman <pwiseman@gmail.com>]

I have a menu item, 'sudo gedit', which has worked for years but just
stopped working (on a current testing system).  Running from the
command line, I get:

No protocol specified

(gedit:30084): Gtk-WARNING **: cannot open display: :0.0

as I do when I try to run gedit in a root terminal.  I hope this is
not another example of putting misplaced security concerns ahead of
functionality.  It's not a problem with my sudoers file as another
menu item, which runs 'sudo aptitude' in a terminal, works.  How can I
run gedit as root?

Patrick