A dimension-free Hermite–Hadamard inequality via gradient estimates for the torsion function

Abstract

Let Ω ⊂ R n be a convex domain and let f : Ω → R be a subharmonic function, ∆f ≥ 0, which satisfies f ≥ 0 on the boundary ∂Ω.ThenOur proof is based on a new gradient estimate for the torsion function, ∆u = −1 with Dirichlet boundary conditions, which is of independent interest.2010 Mathematics Subject Classification. 26B25, 28A75, 31A05, 31B05, 35B50.

Improved Bounds for Hermite–Hadamard Inequalities in Higher Dimensions

“…We observe that, as n tends to infinity, ω 1/n n √ n → √ 2πe. We also note a construction from [14] which shows that the constant in Theorem 3 is at most a factor √ 2 from optimal in high dimensions.…”

“…Implicitly, this also gives a characterization of extremizing functions (via the Green's function). Jianfeng Lu and the last author [14] used this proposition in combination with a gradient estimate for the torsion function to show that the best constant in (2) is uniformly bounded in the dimension. We will follow a similar strategy to obtain an improved bound for the optimal constant in (2).…”

“…There are two different approaches: we can interpret it as the maximum lifetime of Brownian motion inside a domain of given measure or we can interpret it as the solution of a partial differential equation to which Talenti's theorem [23] can be applied. In both cases, we end up with a standard isoperimetric estimate [23] (that was also used in [14])…”

“…This inequality is rather elementary and has been refined in many ways -we refer to the monograph of Dragomir & Pearce [7]. However, there is relatively little work outside of the one-dimensional case; we refer to [3,4,14,15,16,17,18,21]. The strongest possible statement that one could hope for is, for convex functions f : Ω → R defined on convex domains Ω ⊂ R n , 1…”

Hot spots in convex domains are in the tips (up to an inradius)

“…Standard estimates suggest a dichotomy: roughly const · ε of Brownian particles 'escape' (in the sense of not hitting the wall but hitting the boundary ∂Ω instead, roughly 1 − const · ε hit the wall before hitting the boundary. The ones hitting the wall can be analyzed fairly completely, there is a fairly straightforward computation exploiting the reflection principle that is, for example, carried out in [20]. However, the typical hitting time for these particles is rather small and on the scale ε 2 T ≪ ε, this means that the factor e µ1T does not contribute very much.…”