Two Volume Product Inequalities and Their Applications

Stancu, Alina

doi:10.4153/cmb-2009-049-0

Cited by 22 publications

(21 citation statements)

References 27 publications

(22 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This conjecture has been verified among unconditional bodies [16,10] (which correspond to symmetries around coordinate hyperplanes) and this result has been extended to more general symmetries [2]. On the other hand, [13] (improving earlier work, [19]), showed that if a convex body K has a boundary point with positive generalized Gauss curvature, then K cannot minimize the volume product.…”

Section: Introductionmentioning

confidence: 80%

A Simplicial Polytope That Maximizes the Isotropic Constant Must Be a Simplex

Rademacher

2015

Mathematika

View full text Add to dashboard Cite

The isotropic constant L K is an affine-invariant measure of the spread of a convex body K., where A(K) is the covariance matrix of the uniform distribution on K. It is an outstanding open problem to find a tight asymptotic upper bound of the isotropic constant as a function of the dimension. It has been conjectured that there is a universal constant upper bound. The conjecture is known to be true for several families of bodies, in particular, highly symmetric bodies such as bodies having an unconditional basis. It is also known that maximizers cannot be smooth.In this work we study the gap between smooth bodies and highly symmetric bodies by showing progress towards reducing to a highly symmetric case among non-smooth bodies. More precisely, we study the set of maximizers among simplicial polytopes and we show that if a simplicial polytope K is a maximizer of the isotropic constant among d-dimensional convex bodies, then when K is put in isotropic position it is symmetric around any hyperplane spanned by a (d − 2)-dimensional face and the origin. By a result of Campi, Colesanti and Gronchi, this implies that a simplicial polytope that maximizes the isotropic constant must be a simplex.

show abstract

Section: Introductionmentioning

confidence: 80%

A Simplicial Polytope That Maximizes the Isotropic Constant Must Be a Simplex

Rademacher

2015

Mathematika

View full text Add to dashboard Cite

show abstract

“…The case of n = 2 was proved by Mahler [23]. It was also proved in several special cases, like, e.g., unconditional bodies [42,26,35,4], zonoids [34,13,6], bodies of revolution [29] and bodies with some positive curvature assumption [43,37,12]. An isomorphic version of the conjectures was proved by Bourgain and Milman [8]: there is a universal constant c > 0 such that P(K) ≥ c n P(B n 2 ); see also different proofs in [22,31,15].…”

Section: Introductionmentioning

confidence: 97%

Stability of the reverse Blaschke-Santaló inequality for unconditional convex bodies

Kim

Zvavitch

2014

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Mahler's conjecture asks whether the cube is a minimizer for the volume product of a body and its polar in the class of symmetric convex bodies in R n . The inequality corresponding to the conjecture is sometimes called the reverse Blaschke-Santaló inequality. The conjecture is known to be true in R 2 and for several special cases. In the class of unconditional convex bodies, Saint-Raymond confirmed the conjecture, and Meyer and Reisner, independently, characterized the equality case. In this paper we present a stability version of these results and also show that any symmetric convex body, which is sufficiently close to an unconditional body, satisfies the reverse Blaschke-Santaló inequality.

show abstract

“…Within the last few years, a substantial amount of research was devoted to investigate applications of geometric flows to different areas of mathematics. In particular, there are several major contributions of geometric flows to convex geometry: a proof of the affine isoperimetric inequality by Andrews using the affine normal flow [4], obtaining the necessary and sufficient conditions for the existence of a solution to the discrete L 0 -Minkowski problem using crystalline curvature flow by Stancu [70,71,73] and independently by Andrews [8], an application of the affine normal flow to the regularity of minimizers of Mahler volume by Stancu [72], and obtaining quermassintegral inequalities for k-convex star-shaped domains using a family of expanding flows [33]. To state our stability result, we recall the Banach-Mazur distance.…”

Section: Introductionmentioning

confidence: 99%

The planar Busemann-Petty centroid inequality and its stability

Ivaki

2015

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. In [Centro-affine invariants for smooth convex bodies, Int. Math. Res. Notices. doi: 10.1093/imrn/rnr110, 2011] Stancu introduced a family of centro-affine normal flows, p-flow, for 1 ≤ p < ∞. Here we investigate the asymptotic behavior of the planar p-flow for p = ∞ in the class of smooth, origin-symmetric convex bodies. First, we prove that the ∞-flow evolves suitably normalized origin-symmetric solutions to the unit disk in the Hausdorff metric, modulo SL(2). Second, using the ∞-flow and a Harnack estimate for this flow, we prove a stability version of the planar Busemann-Petty centroid inequality in the Banach-Mazur distance. Third, we prove that the convergence of normalized solutions in the Hausdorff metric can be improved to convergence in the C ∞ topology.

show abstract

Two Volume Product Inequalities and Their Applications

Cited by 22 publications

References 27 publications

A Simplicial Polytope That Maximizes the Isotropic Constant Must Be a Simplex

A Simplicial Polytope That Maximizes the Isotropic Constant Must Be a Simplex

Stability of the reverse Blaschke-Santaló inequality for unconditional convex bodies

The planar Busemann-Petty centroid inequality and its stability

Contact Info

Product

Resources

About