TY - JOUR

T1 - On the Space of Convex Figures

AU - Makeev, V. V.

AU - Netsvetaev, N. Yu.

N1 - Makeev, V.V., Netsvetaev, N.Y. On the Space of Convex Figures. J Math Sci 212, 533–535 (2016). https://doi.org/10.1007/s10958-016-2682-8

PY - 2016

Y1 - 2016

N2 - Let T be the set of convex bodies in ℝk, and let T be the set of their similarity classes. If k = 2, then F is written instead of T. Ametric d on T is defined by setting d({K1}, {K2}) = inf{log(b/a)} for the classes {K1}, {K2} ∈ T of convex bodies K1 and K2, where a and b are positive reals such that there is a similarity transformation A with aA(K1) ⊂ K2 ⊂ bA(K1). Let D2 be the planar unit disk. If x > 0, then Fx denotes the set of planar convex figures K in F with d({D2}, {K}) ≥ x. In addition, the sets T and F are equipped with the usual Hausdorff metric. It is proved that if y > log(sec(π/n)) ≥ x for a certain integer n greater than 2, then no mapping Fx → Fy is SO(2)-equivariant. Let Mk denote the space of k-dimensional convex polyhedra and let Mk(n) ⊂ Mk be the space of polyhedra with at most n hyperfaces (vertices). It is proved that there are no SO(k)-equivariant continuous mappings Mk(n + k) → Mk(n). Let T s be the closed subspace of T formed by centrally symmetric bodies. Let Tx denote the closed subspace of T formed by the bodies K with d(T s, {K}) ≥ x > 0. It is proved that for each positive y there exists a positive x such that no mapping Tx → Ty is SO(k)-equivariant.

AB - Let T be the set of convex bodies in ℝk, and let T be the set of their similarity classes. If k = 2, then F is written instead of T. Ametric d on T is defined by setting d({K1}, {K2}) = inf{log(b/a)} for the classes {K1}, {K2} ∈ T of convex bodies K1 and K2, where a and b are positive reals such that there is a similarity transformation A with aA(K1) ⊂ K2 ⊂ bA(K1). Let D2 be the planar unit disk. If x > 0, then Fx denotes the set of planar convex figures K in F with d({D2}, {K}) ≥ x. In addition, the sets T and F are equipped with the usual Hausdorff metric. It is proved that if y > log(sec(π/n)) ≥ x for a certain integer n greater than 2, then no mapping Fx → Fy is SO(2)-equivariant. Let Mk denote the space of k-dimensional convex polyhedra and let Mk(n) ⊂ Mk be the space of polyhedra with at most n hyperfaces (vertices). It is proved that there are no SO(k)-equivariant continuous mappings Mk(n + k) → Mk(n). Let T s be the closed subspace of T formed by centrally symmetric bodies. Let Tx denote the closed subspace of T formed by the bodies K with d(T s, {K}) ≥ x > 0. It is proved that for each positive y there exists a positive x such that no mapping Tx → Ty is SO(k)-equivariant.

KW - Vector Bundle

KW - Convex Body

KW - Closed Subspace

KW - Similarity Transformation

KW - Equivariant Mapping

UR - https://link.springer.com/article/10.1007/s10958-016-2682-8

M3 - Article

VL - 212

SP - 533

EP - 535

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 5

ER -