Convex sets can have interior hot spots

A homogeneous, insulated object with a non-uniform initial temperature will eventually reach thermal equilibrium. Which point in the object takes the longest to reach this equilibrium? Intuition tells us that points “in the middle” of the material will generically reach this equilibrium temperature faster, and points “far from the middle” (i.e. at the boundary) should take more time to reach this temperature. The hot spots conjecture is a formalization of this intuition.

Rauch initially conjectured that points attaining the maximum temperature in the material would approach the boundary as time goes to infinity. Burdzy and Werner later disproved the conjecture for planar domains with holes. The general consensus, however, was that the conjecture should still hold for convex sets of all dimensions.

This talk will draw inspiration from a recurrent theme in convex analysis: most dimension-free results in convex analysis have a natural log-concave extension. We will motivate and construct the log-concave analog of the Hot Spots conjecture, and then disprove it. Using this log-concave construction, we will argue that the hot spots conjecture for convex sets is false in high enough dimensions.