论文标题
在一对凸形机构的纯属上
On an equichordal property of a pair of convex bodies
论文作者
论文摘要
令$ d \ ge 2 $，让$ k $和$ l $为$ {\ mathbb r^d} $中的两个凸体，这样$ l \ subset \ subset \ textrm {int} \，k $，$ l $的边界不包含一个细分。如果$ k $和$ l $满足$（d+1）$ - equichordal属性，即，对于支持$ l $的边界的任何行$ l $以及$ k $ l $ k $ l $ l $，$ k $ of $ k $的交叉点$ \ {ζ_ {pm} \} $ ζ_+）+\ textrm {dist}^{d+1}（l \ cap l，ζ_-）=2σ^{d+1} $$保持，其中常数$σ$独立于$ l $，它是否遵循$ k $和$ k $和$ l $是同心的euclidean kalls？我们证明，如果$ k $和$ l $具有$ c^2 $ - 平滑边界，而$ l $是一场革命的机构，那么$ k $和$ l $是同心的欧几里得球。
Let $d\ge 2$ and let $K$ and $L$ be two convex bodies in ${\mathbb R^d}$ such that $L\subset \textrm{int}\,K$ and the boundary of $L$ does not contain a segment. If $K$ and $L$ satisfy the $(d+1)$-equichordal property, i.e., for any line $l$ supporting the boundary of $L$ and the points $\{ζ_{\pm}\}$ of the intersection of the boundary of $K$ with $l$, $$ \textrm{dist}^{d+1}(L\cap l, ζ_+)+\textrm{dist}^{d+1}(L\cap l, ζ_-)=2σ^{d+1} $$ holds, where the constant $σ$ is independent of $l$, does it follow that $K$ and $L$ are concentric Euclidean balls? We prove that if $K$ and $L$ have $C^2$-smooth boundaries and $L$ is a body of revolution, then $K$ and $L$ are concentric Euclidean balls.
