Dimitrios Liappis addresses the challenge of resizing the root EBS volume on Linux-based Amazon AMIs, particularly for Debian instances, where increasing the default EBS size does not automatically expand the root filesystem. This issue arises because Debian AMIs lack certain tools like growpart that facilitate automatic resizing, unlike CentOS AMIs which include the necessary packages. To resolve this, Liappis suggests using the 'parted' tool with an init script to resize the root partition non-interactively upon launching a new instance. This script detects discrepancies between the EBS volume and partition sizes, resizes the partition using a specific command, and then enables the filesystem to expand automatically on reboot with cloud-init tools. He also notes that other AMIs, such as CentOS6 and SLES11.x, face similar issues and recommends using growpart where available. The solution involves creating a custom AMI that includes an init script to automate these steps, ensuring the full use of the allocated EBS volume.