The log-Brunn–Minkowski inequality

Abstract

We prove the log-Brunn-Minkowski conjecture for convex bodies with symmetries to n independent hyperplanes, and discuss the equality case and the uniqueness of the solution of the related case of the logarithmic Minkowski problem. We also clarify a small gap in the known argument classifying the equality case of the log-Brunn-Minkowski conjecture for unconditional convex bodies. *

On the uniqueness ofLp-Minkowski problems: The constant p-curvature case inR3

“…Hence, to study the uniqueness of the L p -Minkowski problem for p < 1, one needs to impose more conditions on the convex body K or on the function f . In the case of n = 1, for 0 ≤ p < 1, Böröczky-Lutwak-Yang-Zhang [6] obtained the analogous inequalities to (1.4) for origin-symmetric convex bodies, which further implies the uniqueness under these assumptions. When p = 0, the uniqueness was due to Gage [17] within the class of origin-symmetric plane convex bodies that are also smooth and have positive curvature; while when the plane convex bodies are polytopes, the uniqueness was obtained by Stancu [42].…”

“…When p = 0, the uniqueness was due to Gage [17] within the class of origin-symmetric plane convex bodies that are also smooth and have positive curvature; while when the plane convex bodies are polytopes, the uniqueness was obtained by Stancu [42]. As mentioned in [6]: "For plane convex bodies that are not origin-symmetric, the uniqueness problem (when 0 ≤ p < 1) remains both open and important. "…”

“…In particular, the following example (as mentioned in [6]) shows that the important Brunn-Minkowski-Firey inequality (1.4), which was a crucial tool to establish the uniqueness for p ≥ 1, does not hold when p < 1 in general.…”

Orlicz mixed affine quermassintegrals

“…Note that ν K (x) exists for almost all x ∈ ∂K with respect to the (n − 1)-dimensional Hausdorff measure on ∂K. The conevolume measure has been investigated widely in various contexts recently, see e.g., [1,2,10,11,18,19,24,25]. If ϕ(t) = t p , p 1, then K + ϕ ε · ϕ L reduces to the L p combination K + p ε · p L, and correspondingly V ϕ (K, L) reduces to the L p mixed volume V p (K, L).…”

The Lp Minkowski problem for polytopes for p<0

“…The reason that uniqueness of solutions to the L p Minkowski problem for p > 1 can be shown is the availability of mixed volume inequalities established by Lutwak [38]. One reason that the L p Minkowski problem becomes challenging when p < 1 is because little is known about the mixed volume inequalities when p < 1 (see, e.g., [6]). In R n , necessary and sufficient conditions for the existence of the solution of the even L p Minkowski problem for the case of 0 < p < 1 was given by Haberl, et al [21].…”