Dyn–Farkhi不等式的范数刚性与等号情形
Norm rigidity and equality cases for the Dyn--Farkhi inequality
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中文总结 AI 辅助
该研究刻画了满足特定$K$-豪斯多夫距离平方次可加性的原点对称凸体$K$，证明其当且仅当为中心在原点的椭圆时成立，并进一步给出等号成立的条件。
中文摘要 AI 辅助
针对关于原点对称的凸体$K\subset\mathbb{R}^2$及非空集合$S\subset\mathbb{R}^2$，我们研究从集合$S$的凸包出发的$K$-豪斯多夫距离，其定义为$d^{(K)}(S):=\sup_{x\in \text{conv}(S)}\inf_{s\in S}\\|x-s\\|_K$，其中$\\|\cdot \\|_K$是以$K$为闭单位球的范数。我们刻画满足对所有非空紧集$A,B\subset\mathbb{R}^2$均有$d^{(K)}(A+B)^2\leq d^{(K)}(A)^2+d^{(K)}(B)^2$的原点对称凸体$K$，解决该问题并证明该性质成立当且仅当$K$是中心在原点的椭圆，随后刻画$K$为椭圆时该不等式等号成立的条件。
英文摘要
For a convex body $K\subset\mathbb{R}^2$ that is symmetric with respect to the origin, and for a nonempty set $S\subset\mathbb{R}^2$, we study the $K$-Hausdorff distance from convex hull, defined by \begin{align*} d^{(K)}(S):=\sup_{x\in \text{conv}(S)}\inf_{s\in S}\|x-s\|_K, \end{align*} where $\|\cdot \|_K$ is the norm whose closed unit ball is $K$. We consider the problem of characterizing the origin symmetric convex bodies $K$ for which \begin{align*} d^{(K)}(A+B)^2\leq d^{(K)}(A)^2+d^{(K)}(B)^2 \end{align*} holds for all nonempty compact $A,B\subset\mathbb{R}^2$. We solve this problem, proving that this property holds if and only if $K$ is an ellipse centered at $0$. We then characterize the conditions for equality for this bound when $K$ is an ellipse.
