Geodesic
An asymmetric convex body with maximal sections of constant volume
Nazarov, Fedor1 ; Ryabogin, Dmitry1 ; Zvavitch, Artem1
1 Department of Mathematics, Kent State University, Kent, Ohio 44242
Journal of the American Mathematical Society, Tome 27 (2014) no. 1, pp. 43-68

Voir la notice de l'article provenant de la source American Mathematical Society

We show that in all dimensions $d\ge 3$, there exists an asymmetric convex body of revolution all of whose maximal hyperplane sections have the same volume. This gives the negative answer to the question posed by V. Klee in 1969.
DOI : 10.1090/S0894-0347-2013-00777-8

Nazarov, Fedor 1 ; Ryabogin, Dmitry 1 ; Zvavitch, Artem 1

1 Department of Mathematics, Kent State University, Kent, Ohio 44242 
@article{10_1090_S0894_0347_2013_00777_8,
     author = {Nazarov, Fedor and Ryabogin, Dmitry and Zvavitch, Artem},
     title = {An asymmetric convex body with maximal sections of constant volume},
     journal = {Journal of the American Mathematical Society},
     pages = {43--68},
     publisher = {mathdoc},
     volume = {27},
     number = {1},
     year = {2014},
     doi = {10.1090/S0894-0347-2013-00777-8},
     url = {http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-2013-00777-8/}
}
TY  - JOUR
AU  - Nazarov, Fedor
AU  - Ryabogin, Dmitry
AU  - Zvavitch, Artem
TI  - An asymmetric convex body with maximal sections of constant volume
JO  - Journal of the American Mathematical Society
PY  - 2014
SP  - 43
EP  - 68
VL  - 27
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-2013-00777-8/
DO  - 10.1090/S0894-0347-2013-00777-8
ID  - 10_1090_S0894_0347_2013_00777_8
ER  - 
%0 Journal Article
%A Nazarov, Fedor
%A Ryabogin, Dmitry
%A Zvavitch, Artem
%T An asymmetric convex body with maximal sections of constant volume
%J Journal of the American Mathematical Society
%D 2014
%P 43-68
%V 27
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-2013-00777-8/
%R 10.1090/S0894-0347-2013-00777-8
%F 10_1090_S0894_0347_2013_00777_8
Nazarov, Fedor; Ryabogin, Dmitry; Zvavitch, Artem. An asymmetric convex body with maximal sections of constant volume. Journal of the American Mathematical Society, Tome 27 (2014) no. 1, pp. 43-68. doi: 10.1090/S0894-0347-2013-00777-8

[1] Gardner, Richard J. Geometric tomography 2006

[2] Gardner, Richard J., Ryabogin, Dmitry, Yaskin, Vlad, Zvavitch, Artem A problem of Klee on inner section functions of convex bodies J. Differential Geom. 2012 261 279

[3] Goodey, Paul, Schneider, Rolf, Weil, Wolfgang On the determination of convex bodies by projection functions Bull. London Math. Soc. 1997 82 88

[4] Helgason, Sigurdur The Radon transform 1999

[5] Klee, Victor Research Problems: Is a Body Spherical If Its 𝐻𝐴-Measurements are Constant? Amer. Math. Monthly 1969 539 542

[6] Matouå¡Ek, Jiå™Ã­ Using the Borsuk-Ulam theorem 2003

[7] Nazarov, Fedor, Ryabogin, Dmitry, Zvavitch, Artem Non-uniqueness of convex bodies with prescribed volumes of sections and projections Mathematika 2013 213 221

[8] Ryabogin, Dmitry, Yaskin, Vlad On counterexamples in questions of unique determination of convex bodies Proc. Amer. Math. Soc. 2013 2869 2874

Cité par Sources :