On the space of convex figures

V. V. Makeev   ; N. Yu. Netsvetaev

Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 12, Tome 415 (2013), pp. 24-28

Résumé

Let $T$ be the set of convex bodies in $\mathbb R^k$, and let $\mathcal T$ be the set of classes of similar bodies in $T$. We write $F$ for $T$ in the case $k=2$. Define a metric $d$ on $\mathcal T$ by setting for classes $\{K_1\},\{K_2\}$ (from $\mathcal T$, of convex bodies $K_1,K_2$) $d(\{K_1\},\{K_2\}) =\inf\{\ln(b/a)\}$, where $a$ and $b$ are positive reals such that there is a similarity transformation $A$ with $aA(K_1)\subset K_2\subset bA(K_1)$. Let $D_2$ be a planar unit disk. If $x>0$, we denote by $F_x$ the set of the planar convex figures $K$ in $F$ with $d(\{D_2\},\{K\})\ge x$. We also equip the sets $T$ and $F$ with the usual Hausdorff metric. We prove that if $y>\ln(\operatorname{sec}(\pi/n))\ge x$ for some integer $n>2$, then no mapping $F_x\to F_y$ is $\operatorname{SO}(2)$-equivariant. Let $M_k (n)$ be the space of $k$-dimensional convex polyhedra with at most $n$ hyperfaces (vertices), and let $M_k$ denote the space of $k$-dimensional convex polyhedra. We prove that there are no $\operatorname{SO}(2)$-equivariant continuous mappings $M_k(n+k)\to M_k(n)$. Let $T^s$ be the closed subspace of $T$ formed by centrally symmetric bodies. Let $T_x$ denote the closed subspace of $T$ formed by the bodies $K$ with $d(T^s,\{K\})\ge x>0$. We prove that for every $y>0$ there exists an $x>0$ such that no mapping $T_x\to T_y$ is $\operatorname{SO}(2)$-equivariant.

Bibliographie

[1] B. Gryunbaum, Etyudy po kombinatornoi geometrii i teorii vypuklykh tel, Nauka, M., 1971 | MR

[2] V. V. Makeev, “Nekotorye ekstremalnye zadachi dlya vektornykh rassloenii”, Algebra i analiz, 19:2 (2007), 131–155 | MR | Zbl

[3] V. V. Makeev, “O nekotorykh kombinatorno-geometricheskikh zadachakh dlya vektornykh rassloenii”, Algebra i analiz, 14:6 (2002), 169–191 | MR | Zbl