Page 140 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 140
VEX BODIES - APPROXIMATION AND SECTIONS (MS-61)
The Golden ratio and high dimensional mean inequalities
Bernardo González Merino, bgmerino@um.es
Universidad de Murcia, Spain
Coauthors: René Brandenberg, Katherina von Dichter
The classical inequalities between means state that
a−1 + b−1 −1 √ a+b
min{a, b} ≤ ≤ ab ≤ ≤ max{a, b}, (1)
22
for any a, b > 0, with equality if and only if a = b.
One can naturally extend (1) considering means of n-dimensional compact, convex sets. No-
tice that means of K and −K are commonly known as symmetrizations of K. In this context,
we will show that in even dimensions, if K has a large Minkowski asymmetry, then the corre-
sponding inequalities between the symmetrizations of K can no longer be optimal. Especially
in the planar case, we compute that the range of asymmetries of K for which the inequalities
between the symmetrizations of K can be optimal is [1, φ], where φ is the Golden ratio. Indeed,
we introduce the Golden House, (up to linear transformations) the only Minkowski centered set
with asymmetry φ such that the symmetrizations of K are successively optimally contained in
each other.
Functional John and Löwner Ellipsoids
Grigory Ivanov, grimivanov@gmail.com
Institute of Science and Technology Austria (IST Austria), Austria
In this talk, we will speak about functional analogs of the John and Löwner ellipsoids for log-
concave functions. We will discuss the existence and uniqueness results for these objects, their
general properties such as volume ratio, containment, John’s condition, etc. We will note the
difference between the behavior of convex sets and log-concave functions concerning our prob-
lems, and highlight a number of open problems.
A solution to some problems of Conway and Guy on monostable
polyhedra
Zsolt Lángi, zlangi@math.bme.hu
Budapest University of Technology and Economics, Hungary
A convex polyhedron is called monostable if it can rest in stable position only on one of its
faces. In this talk we investigate three questions of Conway, regarding monostable polyhedra,
from the open problem book of Croft, Falconer and Guy (Unsolved Problems in Geometry,
Springer, New York, 1991), which first appeared in the literature in a 1969 paper. In this talk we
answer two of these problems. The main tool of our proof is a general theorem describing ap-
proximations of smooth convex bodies by convex polyhedra in terms of their static equilibrium
points.
138
The Golden ratio and high dimensional mean inequalities
Bernardo González Merino, bgmerino@um.es
Universidad de Murcia, Spain
Coauthors: René Brandenberg, Katherina von Dichter
The classical inequalities between means state that
a−1 + b−1 −1 √ a+b
min{a, b} ≤ ≤ ab ≤ ≤ max{a, b}, (1)
22
for any a, b > 0, with equality if and only if a = b.
One can naturally extend (1) considering means of n-dimensional compact, convex sets. No-
tice that means of K and −K are commonly known as symmetrizations of K. In this context,
we will show that in even dimensions, if K has a large Minkowski asymmetry, then the corre-
sponding inequalities between the symmetrizations of K can no longer be optimal. Especially
in the planar case, we compute that the range of asymmetries of K for which the inequalities
between the symmetrizations of K can be optimal is [1, φ], where φ is the Golden ratio. Indeed,
we introduce the Golden House, (up to linear transformations) the only Minkowski centered set
with asymmetry φ such that the symmetrizations of K are successively optimally contained in
each other.
Functional John and Löwner Ellipsoids
Grigory Ivanov, grimivanov@gmail.com
Institute of Science and Technology Austria (IST Austria), Austria
In this talk, we will speak about functional analogs of the John and Löwner ellipsoids for log-
concave functions. We will discuss the existence and uniqueness results for these objects, their
general properties such as volume ratio, containment, John’s condition, etc. We will note the
difference between the behavior of convex sets and log-concave functions concerning our prob-
lems, and highlight a number of open problems.
A solution to some problems of Conway and Guy on monostable
polyhedra
Zsolt Lángi, zlangi@math.bme.hu
Budapest University of Technology and Economics, Hungary
A convex polyhedron is called monostable if it can rest in stable position only on one of its
faces. In this talk we investigate three questions of Conway, regarding monostable polyhedra,
from the open problem book of Croft, Falconer and Guy (Unsolved Problems in Geometry,
Springer, New York, 1991), which first appeared in the literature in a 1969 paper. In this talk we
answer two of these problems. The main tool of our proof is a general theorem describing ap-
proximations of smooth convex bodies by convex polyhedra in terms of their static equilibrium
points.
138
