Page 141 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 141
CONVEX BODIES - APPROXIMATION AND SECTIONS (MS-61)
On the complex hypothesis of Banach
Luis Montejano, luismontej@gmail.com
Instituto de Matematicas, Mexico
The following is known as the geometric hypothesis of Banach: let V be an m-dimensional
Banach space (over the real or the complex numbers) with unit ball B and suppose all n-
dimensional subspaces of V are isometric (all the n-sections of B are affinely equivalent). In
1932, Banach conjectured that under this hypothesis V is a Hilbert space (the boundary of B is
an ellipsoid). Gromow proved in 1967 that the conjecture is true for n=even and Dvoretzky and
V. Milman derived the same conclusion under the hypothesis n=infinity. We prove this conjec-
ture for n = 4k + 1, with the possible exception of V a real Banach space and n = 133. [G.Bor,
L.Hernandez-Lamoneda, V. Jiménez and L. Montejano. To appear Geometry & Topology] for
the real case and [J. Bracho, L. Montejano, submitted to J. of Convex Analysis] for the complex
case.
The ingredients of the proof are classical homotopic theory, irreducible representations of
the orthogonal group and convex geometry. For the complex case, suppose B is a convex body
contained in complex space Cn+1, with the property that all its complex n-sections through
the origin are complex affinity equivalent to a fixed complex n-dimensional body K. Studying
the topology of the complex fibre bundle SU (n)− > SU (n − 1)− > S2n+1, it is possible to
prove that if n=even, then K must be a ball and using homotopical properties of the irreducible
representations we prove that if n = 4 + 1 then K must be a body of revolution. Finally, we
prove, using convex geometry and topology that, if this is the case, then there must be a section
of B which is an complex ellipsoid and consequently B must be also a complex ellipsoid.
Pea Bodies of Constant Width
Deborah Oliveros, deboliveros@gmail.com
Universidad Nacional Autónoma de México (UNAM), Mexico
Besides the two Meissner solids, the obvious constant width bodies of revolution, and the Meiss-
ner polyhedra, there are few concrete examples of bodies of constant width or a concrete finite
procedure to construct them. In this talk, we will describe an infinite family of 3-dimensional
bodies of constant width obtained from the Reuleaux Tetrahedron by replacing a small neigh-
borhood of all six edges with sections of an envelope of spheres using the classical notion of
confocal quadrics. This family includes Meissner solids as well as one with tetrahedral symme-
try. (Joint work with I. Arelio and L. Montejano).
A cap covering theorem
Alexandr Polyanskii, alexander.polyanskii@yandex.ru
MIPT, Russian Federation
A cap of spherical radius α on a unit d-sphere S is the set of points within spherical distance α
from a given point on the sphere. Let F be a finite set of caps lying on S. We prove that if no
hyperplane through the center of S divides F into two non-empty subsets without intersecting
any cap in F, then there is a cap of radius equal to the sum of radii of all caps in F covering all
139
On the complex hypothesis of Banach
Luis Montejano, luismontej@gmail.com
Instituto de Matematicas, Mexico
The following is known as the geometric hypothesis of Banach: let V be an m-dimensional
Banach space (over the real or the complex numbers) with unit ball B and suppose all n-
dimensional subspaces of V are isometric (all the n-sections of B are affinely equivalent). In
1932, Banach conjectured that under this hypothesis V is a Hilbert space (the boundary of B is
an ellipsoid). Gromow proved in 1967 that the conjecture is true for n=even and Dvoretzky and
V. Milman derived the same conclusion under the hypothesis n=infinity. We prove this conjec-
ture for n = 4k + 1, with the possible exception of V a real Banach space and n = 133. [G.Bor,
L.Hernandez-Lamoneda, V. Jiménez and L. Montejano. To appear Geometry & Topology] for
the real case and [J. Bracho, L. Montejano, submitted to J. of Convex Analysis] for the complex
case.
The ingredients of the proof are classical homotopic theory, irreducible representations of
the orthogonal group and convex geometry. For the complex case, suppose B is a convex body
contained in complex space Cn+1, with the property that all its complex n-sections through
the origin are complex affinity equivalent to a fixed complex n-dimensional body K. Studying
the topology of the complex fibre bundle SU (n)− > SU (n − 1)− > S2n+1, it is possible to
prove that if n=even, then K must be a ball and using homotopical properties of the irreducible
representations we prove that if n = 4 + 1 then K must be a body of revolution. Finally, we
prove, using convex geometry and topology that, if this is the case, then there must be a section
of B which is an complex ellipsoid and consequently B must be also a complex ellipsoid.
Pea Bodies of Constant Width
Deborah Oliveros, deboliveros@gmail.com
Universidad Nacional Autónoma de México (UNAM), Mexico
Besides the two Meissner solids, the obvious constant width bodies of revolution, and the Meiss-
ner polyhedra, there are few concrete examples of bodies of constant width or a concrete finite
procedure to construct them. In this talk, we will describe an infinite family of 3-dimensional
bodies of constant width obtained from the Reuleaux Tetrahedron by replacing a small neigh-
borhood of all six edges with sections of an envelope of spheres using the classical notion of
confocal quadrics. This family includes Meissner solids as well as one with tetrahedral symme-
try. (Joint work with I. Arelio and L. Montejano).
A cap covering theorem
Alexandr Polyanskii, alexander.polyanskii@yandex.ru
MIPT, Russian Federation
A cap of spherical radius α on a unit d-sphere S is the set of points within spherical distance α
from a given point on the sphere. Let F be a finite set of caps lying on S. We prove that if no
hyperplane through the center of S divides F into two non-empty subsets without intersecting
any cap in F, then there is a cap of radius equal to the sum of radii of all caps in F covering all
139
