Example: I'd like to have www.mydomain.com/blog
http://www.tumblr.com/docs/en/custom_domains
In brief, you would set up a custom domain name, and then if necessary set a redirect so that "blog.mydomain.com" points to "mydomain.com/blog".
Im not sure if this would have any averse effect on the site's SEO though. How did you deal with that? Did you do a 301 redirect for everything?
Example blog.mydomain.com/article1 to www.mydomain.com/blog/article1
Example: I'd like to have www.mydomain.com/blog