论文标题
对凸组的极性相关性的拓扑见解
A topological insight into the polar involution of convex sets
论文作者
论文摘要
用$ \ MATHCAL {K} _0^n $表示所有闭合凸的家族$ a \ subset \ subset \ mathbb {r}^n $ contement in \ mathbb r^n $。对于$ a \ in \ Mathcal {k} _0^n，其极性集由$ a^\ circ。$表示，在本文中，我们调查了Polar映射$ A \ to $（\ Mathcal {k nathcal {k} _0^_0^n，d_ aw {aw} $ ant $ dd_ $ de at at的拓扑性质。我们证明$（\ Mathcal {k} _0^n，d_ {aw}）$对Hilbert Cube $ q = \ prod_ {i = 1}^{\ infty} [ - 1,1] $，极性映射是基于标准的Free $ progient $ $ q， $σ（x）= - x $ for Q. $ inq。$。我们还证明，在$ \ Mathcal k^n_0 $（也称为二元）上的包含 - 逆转限制中，那些和只有独特固定点的二重性和唯一的固定点在极地上与极性映射相结合，并且它们可以表征为所有映射$ f：\ nath n n n n v：\ k}^0^k}^k}^k}^k}^k} {k} {k} {k {k {k {k {k} {k {k {k {k {k {k {k {k {k} { \ Mathcal {k} _0^n $的表格$ f（a）= t（a^{\ circ}）$，带有$ t $的$ \ mathbb r^n $的正确定线性同构。
Denote by $\mathcal{K}_0^n$ the family of all closed convex sets $A\subset\mathbb{R}^n$ containing the origin $0\in\mathbb R^n$. For $A\in\mathcal{K}_0^n,$ its polar set is denoted by $A^\circ.$ In this paper, we investigate the topological nature of the polar mapping $A\to A^\circ$ on $(\mathcal{K}_0^n, d_{AW})$, where $d_{AW}$ denotes the Attouch-Wets metric. We prove that $(\mathcal{K}_0^n, d_{AW})$ is homeomorphic to the Hilbert cube $Q=\prod_{i=1}^{\infty}[-1,1]$ and the polar mapping is topologically conjugate with the standard based-free involution $σ:Q\rightarrow Q,$ defined by $σ(x)=-x$ for all $x\in Q.$ We also prove that among the inclusion-reversing involutions on $\mathcal K^n_0$ (also called dualities), those and only those with a unique fixed point are topologically conjugate with the polar mapping, and they can be characterized as all the maps $f:\mathcal{K}_0^n\to \mathcal{K}_0^n$ of the form $f(A)=T(A^{\circ})$, with $T$ a positive definite linear isomorphism of $\mathbb R^n$.
