IF your lease agreement has terms that protect you from such competition, then you can seek to enforce those terms. If it doesn't, then you have only an 'unfair' argument to make with the landlord. Discuss with local counsel your claims and the lease terms, and see if they think it is worth you investing substantial money in negotiation or litigation with the landlord.

Unless you have an agreement with the landlord only to allow one yogurt shop in the shopping center, you do not have any legal recourse. Rights to have exclusivity in shopping center leases are common, and it sounds like the original yogurt shop had one, but they are purely a creature of contract. So if it isn't in your contract, you don't have the rights. I have to assume, though, that the original yogurt shop had such a clause in their lease. I wonder why they aren't enforcing it. If for some reason it's no longer enforceable, maybe you could get the landlord to agree to let you sell yogurt too, and then maybe you'd get the business of couples, families or groups in which one person might want yogurt but the other(s) want something else.

In addition to what my colleagues suggest, I'm thinking of cross-marketing strategies. For instance, maybe you and the yogurt shop could work together. People could try yogurt samples at your store, and your samples at the yogurt store. Put your business card next to the samples. Sometimes being a nice neighbor works wonders!

Unless your lease contains an express covenant against the landlord leasing to a competing business, and the covenant can be interpreted to include selling yogurt, you won't have a valid cause of action against your landlord. There is no implied covenant against a plaza or mall leasing to a competitor of an existing tenant, and when such covenants have been inserted into a lease, they are interpreted rather narrowly in favor of competition. The leading case on this topic, although a bit old now, is the California Supreme Court case Stockton Dry Goods Co. v. Girsh (1951) 36 Cal.2d 677.