I have an install of Exchange 2010 without an Edge Transport Server. The only way I can get it to receive e mail from the internet is by allowing anonymous access to the receive connector but this in turn means the server can be used as a relay. Testing has proved that this is indeed the case.
I've Googled this extensively but can't seem to find any solution, other than installing the Edge Transport role somewhere which unfortunately isn't an option.
I'm hopeful that I'm just missing something obvious - previously I've always used Exchange 2003.