this is what Salesforce does - they have publicly visible shards in the url (na11, na14, eu23, etc) and login.salesforce.com redirects you to your shard when it figures out who you are.
For anyone considering this, reconsider. It bit me many times.
In general, over time, load on some shards will increase while others decrease. Migrating a customer from one shard to another will likley cause a short outage for them, and many bugs down the line when they've bookmarked all kinds of things.
You need to preserve the name<->customer association and maintain another key that you can use to split traffic at the LB in case a customer outgrows their shard. But personally I think it looks hokey and should not be something a customer sees anywhere but perhaps a developer tool or sniffer.
In general, over time, load on some shards will increase while others decrease. Migrating a customer from one shard to another will likley cause a short outage for them, and many bugs down the line when they've bookmarked all kinds of things.