Problem with SSL and MySql disconnection

Hello,
one of our (zen cart italia) user has a strange problem:

He is using SSL, but whenever Zen Cart goes from a https page to a http page, the user is logged out (the same in admin).

He noticed that if he forces every page under ssl, it works (but this is not a good solution, both for performance and because the ssl is on port 30000 and must be written in the url...).

Environment:

Zen Cart v.1.3.7.3 (last .3 is our italian minor version and has no impact on this matter)

Apache: Apache/2.2.2 (Fedora)
Php: PHP/5.1.6
Mysql: 5.0.27

config:

define('HTTP_SERVER', 'http://www.myhost.it');
define('HTTPS_SERVER', 'https://myhost.it:30000');

// Use secure webserver for checkout procedure?
define('ENABLE_SSL', 'true');

define('USE_PCONNECT', 'false'); // use persistent connections?
define('STORE_SESSIONS', ''); // use 'db' for best support, or '' for file-based storage

Any hint about the reason of logging out?

Thanks

s_p_ike:

define('HTTP_SERVER', 'http://www.myhost.it');
define('HTTPS_SERVER', 'https://myhost.it:30000');If you make them both be "www." does it work any better? (ie: add the www to the https URL)

Also, what are the settings in Admin->Configuration->Sessions?