Rethinking about this question (I hope to have understood your problem right, but correct me if I'm wrong), I found an another solution and I think you don't need to run the application as www-data nor to play with permissions. You said that:
Apache's DocumentRoot should be under
chown www-data:www-data /path/to/www
but actually, this isn't fully true: Apache's DocumentRoot must be readable by www-data, but not necessarily owned by it. By default, in Ubuntu Desktop, all files and directories you create are readable by everybody, so you can set your DocumentRoot to a directory inside your home folder, for example:
DocumentRoot /home/user/my-project/my-document-root
In Ubuntu 19.10 and later, the admonition in that article (and in this answer) no longer applies. See WinEunuuchs2Unix's answer as well as this question.
Graphical applications often store settings and other user-specific data in configuration files written inside the user's home folder. The main mechanism applications use to determine what they should use as the user's home folder is the HOME environment variable. (You can inspect it yourself with echo $HOME).
Suppose you're running gedit (a graphical text editor) as root. If you run sudo gedit, HOME will continue to point toward your home directory, even though the program is running as root. Consequently, gedit will write configuration files as root into your home directory. This will sometimes result in the configuration files being owned by root and thus inaccessible to you (when you later run the program as yourself and not as root). This mainly happens when the application has to create a new configuration file. Newly created files, by default, are owned by the user who creates them (who in this case is root, not you).
That's the primary reason why you should run graphical applications with a graphical sudo frontend rather than with straight sudo. In Ubuntu and most of its derivatives (including Xubuntu and Lubuntu), the standard graphical frontend is gksu/gksudo. In Kubuntu it is kdesudo. (It depends on the desktop environment being used.)
If you want to use sudo directly to run a graphical application like gedit, you can run:
sudo -H gedit
The -H flag makes sudo set HOME to point to root's home folder (which is /root).
That still won't automatically handle the ownership of .Xauthority by copying it to a temp folder (this is the other thing that graphical sudo frontends take care of for you). But in the infrequent event that .Xauthority is inaccessible, you'll get an error saying it is, and then you can fix the problem by deleting it (sudo rm ~/.Xauthority), as it's automatically regenerated. Thus, protecting .Xauthority's ownership and permissions is less important than protecting the ownership and permissions of configuration files.
In contrast to a root-owned .Xauthority, when configuration files become owned as root, it's not always as obvious what the problem is (because graphical programs will often run, but not work very well, and output any useful errors to the console). And it's sometimes a bigger hassle to fix, especially if you're in a situation where you want one or more files in your home directory to be owned by someone other than you (because then you cannot fix it simply by recursively chowning all your files back to yourself).
Therefore, sudo (at least without -H) should not be used to run a graphical application unless you are highly familiar with the app's inner workings and know for sure that it does not ever attempt to write any configuration files.
Best Answer
sudo -ishould work in reverse too but you need to state which user you want to run as: