Actually, Terra, you've got it right. The eye is the area of the lowest barometric pressure. The strongest warming in a hurricane is found at upper levels in the eye due to sinking air. In fact, the eye is such a clear region because of the sinking air.

This sinking air is a necessary artifact of hurricanes, in fact; with such strong rising motion in the eyewall (and through the convective mass as well), there must be some compensating sinking motion to maintain balance. Some of this is seen on the outer periphery of the storm, but the majority of this is found within the eye. The sinking causes the air at upper levels to warm -- forced descent is a warming process -- and helps drive the warm-core structure of the cyclone. With all of that heating at upper levels, the strongest response in the hurricane, coincidentally enough, is found at the surface. The response of the storm at the surface is to deepen, leading to increased rising motion and, given favorable surface & upper-air conditions, a feedback cycle upon which the storm can intensify.

The stronger the storm, the stronger the descent within the eye. The need for this descent & the associated warming aloft is the primary reason why you don't ever see the eye contract to a near-zero diameter: if it were to do so, the sinking motion would be cut off, weakening the warming aloft (which in a hurricane is not due to convection), and the pressure would begin to rise. That's why you see eyewall replacement cycles with these storms, as the natural tendency for any eyewall is to contract. The science behind that one is pretty complicated, though (heavy in physical and dynamical meteorology), and I'll leave it for another day.