The Log-Convex Density Conjecture and vertical surface area in warped products

Abstract

Abstract. We examine the vertical component of surface area in the warped product of a Euclidean interval and a fiber manifold with product density. We determine general conditions under which vertical fibers minimize vertical surface area among regions bounding the same volume and use these results to conclude that in many such spaces vertical fibers are isoperimetric. Our main hypothesis is that the surface area of a fiber be a convex function of the volume it bounds. We apply our results in the specific case of R n − {0} realized as the warped product (0, ∞) ×r S n−1 , providing many new examples of densities where spheres about the origin are isoperimetric, including simple densities with finite volume, simple densities that at the origin are neither logconvex nor smooth, and non-simple densities. We also generalize the results of Kolesnikov and Zhdanov on large balls in R n with increasing strictly log-convex simple density. We situate our work in relation to the Log-Convex Density Conjecture of Rosales et al. and the recent work by Morgan, Ritoré, and others on formulating a generalized log-convex density/stable spheres conjecture.

Isoperimetric and Stable Sets for Log-ConcavePerturbations of Gaussian Measures

“…However, in spite of the last advances, the characterization of the solutions has been achieved only for some densities having a special form or a nice behaviour with respect to a certain subgroup of diffeomorphisms. In particular, radial and homogeneous densities are being a focus of attention, see the related works [61], [42], [63], [33], [27], [21], [24], [56], and the references therein.…”

Complemented Brunn–Minkowski inequalities and isoperimetry for homogeneous and non-homogeneous measures

“…Remark 5.16. When B is the Euclidean ball, w 0 and ϕ are continuous and positive on S n−1 and (0, ∞), respectively (so Σ = R n ), and ϕ is not required to be strictly log-convex, the first assertion was obtained by Howe [19]. As our analysis indicates, various subtleties may occur when the positivity assumption on w 0 is removed.…”

Some Weighted Isoperimetric Problems on $${\mathbb {R}}^N _+ $$ with Stable Half Balls Have No Solutions

“…In this section we are interested in criteria for nonexistence and nonradiality of solutions to some weighted isoperimetric problems on R N . More results to these and related questions can be found in the papers [22], [11], [15], [21] and in [19]. Let f, g be two positive functions on R N with g locally integrable and f lower semi-continuous.…”