I know I could easily fill seven days, or more, in Lima; there's actually a lot more there than most people think. First, though, It would be good to know whether your dates, 3-13 Dec., include the days you fly in, and then fly out -- in which case, you have only nine "full" days in the country. In that case, you won't have seven full days for either Lima or Cusco.
As for Lima, I have some interest in Peruvian history and culture, so I'll admit that there are places in the city that interest me, that probably wouldn't interest more general tourists. These include several places of "intermediate" interest; for example, a number of ancient sites, which I find quite fascinating myself, though being adobe structures, in some cases rather eroded, and surrounded by urban Lima, they may lack the "visual spectacle" offered by the well-preserved stone structures of Machu Picchu on their high mountain ridge. (In Lima, Huaca Pucllana and Mateo Salado may offer the visual spectacle that tourists seek.) For Lima, as for so many other cities, if you just want to see two or three of its better-known touristic sites, and say afterwards that you've "seen" Lima, or at least glimpsed it, then two full days (i.e, not counting the day you enter Lima, and the day you leave) may suffice, though barely. But the more interest you have in Lima or Peru, the longer you will want to consider staying.
Also -- my visit to Arequipa was in 2009, so my recollections are obviously far from current; but if it's still the same city that I remember from back then, it's an excellent choice.