A homogeneous, insulated object with a non-uniform initial temperature will eventually reach thermal equilibrium. The Hot Spots conjecture addresses which point in the object takes the longest to reach this equilibrium: Where is the maximum temperature attained as time progresses?
Rauch initially conjectured that points attaining the maximum temperature would approach the boundary for larger times. Burdzy and Werner disproved the conjecture for planar domains with holes. Kawohl, and later Bañuelos-Burdzy, conjectured that the conjecture should still hold for convex sets of all dimensions.
This talk will draw inspiration from a recurrent theme in convex analysis: almost every dimension-free result in convex analysis has 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 show that the hot spots conjecture for convex sets is false in high dimensions.
