Linux – Easiest way to change directory into its own logical volume in LVM2

In theory, you could reduce the size of sda1, increase the size of the extended partition, shift the contents of the extended partition down, then increase the size of the PV on the extended partition and you'd have the extra room. However, the number of possible things that can go wrong there is just astronomical, so I'd recommend either buying a second hard drive (and possibly transferring everything onto it in a more sensible layout, then repartitioning your current drive better) or just making some bind mounts of various bits and pieces out of /home into / to free up a bit more space.

Every time you perform an operation with LVM, by default, the previous metadata is archived in /etc/lvm/archive. You can use vgcfgrestore to restore it, or grab the extends by hand (harder, but lvcreate(8) should cover it).

Edit:

And to make it as easy as possible, I should add that you can find the last backup before your destructive operation by looking at descriptions:

# grep description /etc/lvm/archive/vg01_*
/etc/lvm/archive/vg01_00001.vg:description = "Created before executing 'lvremove -f /dev/vg01/foo'"
/etc/lvm/archive/vg01_00002.vg:description = "Created before executing 'lvremove -f /dev/vg01/bar'"
/etc/lvm/archive/vg01_00003.vg:description = "Created before executing 'lvremove -f /dev/vg01/baz'"

Edit:

The normal allocation policy (default one) will allocate a stripe from the first free PE when there is enough room to do so. If you want to confirm where the LV was allocated, you can look in the archive files, those are perfectly readable by humans.