This recent advisory suggests that one updates his sudo installation. With the subsequent result of being trapped in a mess of obscure, badly documented env_something options, a suggestion of env_check which doesn't work and no way of passing environment variables to the sudo'd process. Great.
Much cursing later it turns out that only this makes sudo tick again:
Defaults env_reset, env_keep="XAUTHORITY DISPLAY"
or, more to my liking in the case of unrestricted sudo,