1) Around the eye 
situated around the orbit, or eye socket.
Around the eyes
Situated around the orbit or eye socket. ... (18 Nov 1997) ... 
around the eye [NHS Direct]
Type: Term Pronunciation: per′ē-ōr′bi-tăl Definitions: 1. Relating to the periorbita. Synonyms: circumorbital
periorbital 1. Situated around the orbit of the eye. 2. Of, relating to, or characterizing the periorbit. 3. Of, relating to, occurring in, or being the tissues surrounding or lining the orbit of the eye; as with periorbital edema.