Page 138 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 138
CONVEX BODIES - APPROXIMATION AND SECTIONS (MS-61)
Extremal sections and local optimization
Gergely Ambrus, ambruge@gmail.com
Alfréd Rényi Institute of Mathematics, Hungary
We will demonstrate how to apply local optimization methods in order to find, or, at least
characterize, maximal or minimal central sections of convex bodies. We are mainly interested
in the special cases when the convex body is the d-dimensional cube, or the d-dimensional
regular simplex. Maximal sections of the former were determined by K. Ball in 1986, while
monotinicity of the volume of diagonal central sections for d ≥ 3 was proven by F. Bartha, F.
Fodor and B. González Merino in 2021. On the other hand, finding minimal central sections
of the regular simplex is still an open question. Among other results, we provide a geometric
characterization for central sections of locally maximal volume of the d-dimensional cube.
Cells in the box and a hyperplane
Imre Bárány, imbarany@gmail.com
Alfréd Rényi Institute of Mathematics, Hungary
It is well known that a line can intersect at most 2n − 1 cells of the n × n chessboard. We
consider the high dimensional version: how many cells of the d-dimensional n × . . . × n box
can a hyperplane intersect? We also prove the lattice analogue of the following well-known
fact. If K, L are convex bodies in Rd and K ⊂ L, then the surface area of K is smaller than
that of L. Joint work with Peter Frankl.
A new look at the Blaschke-Leichtweiss theorem
Károly Bezdek, karoly.bezdek@gmail.com
University of Calgary, Canada
The Blaschke-Leichtweiss theorem (Abh. Math. Sem. Univ. Hamburg 75: 257–284, 2005)
states that the smallest area convex domain of constant width w in the 2-dimensional spherical
space S2 is the spherical Reuleaux triangle for all 0 < w ≤ π . In this paper we extend this result
2
S2, ≤ π
to the family of wide r-disk domains of where 0 < r 2 . Here a wide r-disk domain is an
intersection of spherical disks of radius r with centers contained in their intersection. This gives
a new and short proof for the Blaschke-Leichtweiss theorem. Furthermore, we investigate the
higher dimensional analogue of wide r-disk domains called wide r-ball bodies. In particular,
we determine their minimum spherical width (resp., inradius) in the spherical d-space Sd for all
d ≥ 2. Also, it is shown that any minimum volume wide r-ball body is of constant width r in
Sd, d ≥ 2.
136
Extremal sections and local optimization
Gergely Ambrus, ambruge@gmail.com
Alfréd Rényi Institute of Mathematics, Hungary
We will demonstrate how to apply local optimization methods in order to find, or, at least
characterize, maximal or minimal central sections of convex bodies. We are mainly interested
in the special cases when the convex body is the d-dimensional cube, or the d-dimensional
regular simplex. Maximal sections of the former were determined by K. Ball in 1986, while
monotinicity of the volume of diagonal central sections for d ≥ 3 was proven by F. Bartha, F.
Fodor and B. González Merino in 2021. On the other hand, finding minimal central sections
of the regular simplex is still an open question. Among other results, we provide a geometric
characterization for central sections of locally maximal volume of the d-dimensional cube.
Cells in the box and a hyperplane
Imre Bárány, imbarany@gmail.com
Alfréd Rényi Institute of Mathematics, Hungary
It is well known that a line can intersect at most 2n − 1 cells of the n × n chessboard. We
consider the high dimensional version: how many cells of the d-dimensional n × . . . × n box
can a hyperplane intersect? We also prove the lattice analogue of the following well-known
fact. If K, L are convex bodies in Rd and K ⊂ L, then the surface area of K is smaller than
that of L. Joint work with Peter Frankl.
A new look at the Blaschke-Leichtweiss theorem
Károly Bezdek, karoly.bezdek@gmail.com
University of Calgary, Canada
The Blaschke-Leichtweiss theorem (Abh. Math. Sem. Univ. Hamburg 75: 257–284, 2005)
states that the smallest area convex domain of constant width w in the 2-dimensional spherical
space S2 is the spherical Reuleaux triangle for all 0 < w ≤ π . In this paper we extend this result
2
S2, ≤ π
to the family of wide r-disk domains of where 0 < r 2 . Here a wide r-disk domain is an
intersection of spherical disks of radius r with centers contained in their intersection. This gives
a new and short proof for the Blaschke-Leichtweiss theorem. Furthermore, we investigate the
higher dimensional analogue of wide r-disk domains called wide r-ball bodies. In particular,
we determine their minimum spherical width (resp., inradius) in the spherical d-space Sd for all
d ≥ 2. Also, it is shown that any minimum volume wide r-ball body is of constant width r in
Sd, d ≥ 2.
136
