Shaken Rogers's Theorem for Homothetic Sections

2009 ◽  
Vol 52 (3) ◽  
pp. 403-406
Author(s):  
J. Jerónimo-Castro ◽  
L. Montejano ◽  
E. Morales-Amaya
Keyword(s):  
Convex Bodies ◽  
Strictly Convex

AbstractWe shall prove the following shaken Rogers's theorem for homothetic sections: Let K and L be strictly convex bodies and suppose that for every plane H through the origin we can choose continuously sections of K and L, parallel to H, which are directly homothetic. Then K and L are directly homothetic.

Prescribing Centro-Affine Curvature From One Convex Body to Another

2020 ◽  
Author(s):  
Alina Stancu
Keyword(s):  
Convex Body ◽  
Convex Bodies ◽  
Minkowski Inequality ◽  
Curvature Flow ◽  
Constant Volume ◽  
Convex Hypersurface ◽  
Strictly Convex ◽  
Initial Hypersurface ◽  
Affine Curvature ◽  
Convex Hypersurfaces

Abstract We study a curvature flow on smooth, closed, strictly convex hypersurfaces in $\mathbb{R}^n$, which commutes with the action of $SL(n)$. The flow shrinks the initial hypersurface to a point that, if rescaled to enclose a domain of constant volume, is a smooth, closed, strictly convex hypersurface in $\mathbb{R}^n$ with centro-affine curvature proportional, but not always equal, to the centro-affine curvature of a fixed hypersurface. We outline some consequences of this result for the geometry of convex bodies and the logarithmic Minkowski inequality.

scholarly journals On weak stationarity and weak isotropy of processes of convex bodies and cylinders

2007 ◽  
Vol 39 (04) ◽  
pp. 864-882
Author(s):  
Lars Michael Hoffmann
Keyword(s):  
Convex Bodies ◽  
Nonstationary Process ◽  
Rotation Invariant ◽  
Strictly Convex ◽  
Weak Stationarity ◽  
Local Mean

Generalized local mean normal measures μ z , z ∈ R d , are introduced for a nonstationary process X of convex particles. For processes with strictly convex particles it is then shown that X is weakly stationary and weakly isotropic if and only if μ z is rotation invariant for all z ∈ R d . The paper is concluded by extending this result to processes of cylinders, generalizing Theorem 1 of Schneider (2003).

Nearly all convex bodies are smooth and strictly convex

1987 ◽  
Vol 103 (1) ◽  
pp. 57-62 ◽  
Author(s):  
Tudor Zamfirescu
Keyword(s):  
Convex Bodies ◽  
Strictly Convex

scholarly journals On weak stationarity and weak isotropy of processes of convex bodies and cylinders

2007 ◽  
Vol 39 (4) ◽  
pp. 864-882 ◽  
Author(s):  
Lars Michael Hoffmann
Keyword(s):  
Convex Bodies ◽  
Nonstationary Process ◽  
Rotation Invariant ◽  
Strictly Convex ◽  
Weak Stationarity ◽  
Local Mean

Generalized local mean normal measures μz, z ∈ Rd, are introduced for a nonstationary process X of convex particles. For processes with strictly convex particles it is then shown that X is weakly stationary and weakly isotropic if and only if μz is rotation invariant for all z ∈ Rd. The paper is concluded by extending this result to processes of cylinders, generalizing Theorem 1 of Schneider (2003).

scholarly journals Lens Rigidity in Scattering by Unions of Strictly Convex Bodies in R^2

2020 ◽  
Vol 52 (1) ◽  
pp. 471-480 ◽  
Author(s):  
Lyle Noakes ◽  
Luchezar Stoyanov
Keyword(s):  
Convex Bodies ◽  
Strictly Convex ◽  
Lens Rigidity

Exponential instability for a class of dispersing billiards

1999 ◽  
Vol 19 (1) ◽  
pp. 201-226 ◽  
Author(s):  
LATCHEZAR STOYANOV
Keyword(s):  
Topological Entropy ◽  
Disjoint Union ◽  
Convex Bodies ◽  
Uniform Density ◽  
Minimal Distance ◽  
Billiard Ball ◽  
Strictly Convex ◽  
Exponential Instability ◽  
Dispersing Billiards ◽  
Finite Disjoint Union

The billiard in the exterior of a finite disjoint union $K$ of strictly convex bodies in ${\mathbb R}^d$ with smooth boundaries is considered. The existence of global constants $0 < \delta < 1$ and $C > 0$ is established such that if two billiard trajectories have $n$ successive reflections from the same convex components of $K$, then the distance between their $j$th reflection points is less than $C(\delta^j + \delta^{n-j})$ for a sequence of integers $j$ with uniform density in $\{1,2,\dots,n\}$. Consequently, the billiard ball map (although not continuous in general) is expansive. As applications, an asymptotic of the number of prime closed billiard trajectories is proved which generalizes a result of Morita [Mor], and it is shown that the topological entropy of the billiard flow does not exceed $\log (s-1)/a$, where $s$ is the number of convex components of $K$ and $a$ is the minimal distance between different convex components of $K$.

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