The University of Rochester, Rochester, NY, USA.
Moscow State University,
My primary area of research interest is Minkowski geometry and its applications.
Minkowski geometry is the study of the geometry of finite
dimensional normed linear spaces, where the unit ball can be
chosen to be an arbitrary origin-symmetric convex body. In Minkowski
geometry distance is not "uniform" in all directions. Therefore,
the Pythagoras' Theorem is no longer valid; however, the parallel
axiom is still valid. This field can be located at the intersection
of Convex Geometry, Banach Space Theory, and Finsler Geometry, but
it is also closely related to other areas of mathematics such as
normed spaces, discrete geometry, integral geometry, Fourier
analysis, and symplectic geometry. This field was also enriched by
many results from applied disciplines such as operations research,
optimization, and computational geometry. The theory of Minkowski
geometry has also applications to crystal growth and material
Selected papers and preprints
- On unit balls and isoperimetrices in normed spaces (with H. Martini),
Colloq. Math. , 127 (2012), 133-142.
- On orthogonal chords in normed spaces (with J. Alonso and H. Martini),
Rocky Mountain J. Math 41 (2011), 23-35.
- Estimates on inner and outer radii of unit balls in normed spaces (with H. Martini),
Colloq. Math. 123 (2011), 211-217.
- On isoperimetric inequalities in Minkowski spaces (with H. Martini), Journal of Ineq. & Appl., (2010), Article ID 697954, 18 pp. (electronic).
- A construction of convex figures of constant width (with H. Martini),
Computer Aided Geometric Design 25 (2008), 751-755.
- Extensions of a Bonnesen-style inequality to Minkowski spaces (with H. Martini),
Math. Ineq & Appl., 11 (2008), 739-748.
- On Reuleaux triangles in Minkowski planes (with H. Martini),
Beiträge zur Algebra und Geometrie/Contributions to Algebra and Geometry,
48 (2007), 225-235.
- Some applications of cross-section measures in Minkowski spaces (with H. Martini),
Period. Math. Hungar. 53 (2006), no. 1-2, 185--197.
- On Busemann surface area of the unit ball in Minkowski spaces,
J. Inequal. Pure Appl. Math. 7 (2006), no. 1, Article 19, 10 pp. (electronic).
- The ratio of the length of the unit circle to the area of the unit disc in Minkowski planes,
Proc. Amer. Math. Soc. 133 (2005), no. 4, 1231--1237.
- Some geometric inequalities for the Holmes-Thompson definitions of volume and surface area in Minkowski spaces,
J. Inequal. Pure Appl. Math. 5 (2004), no. 1, Article 17, 10 pp. (electronic).