论文标题
Horocyclic Brunn-Minkowski不平等
Horocyclic Brunn-Minkowski inequality
论文作者
论文摘要
给定了两个非空的子集$ a $ a $ a $ a $ a $ a $ \ mathbb {h}^2 $，我们定义了他们的horocyclic minkowski和参数$λ= 1/2 $作为集合$ [a：b] _ {a：b] _ {1/2} \ subseteeq \ mathbb {h mathbb {h imnond clive $ a $，$ b $中的点。这些horocycle曲线通过双曲线弧长进行了参数化，并且具有参数$ 0 <λ<1 $的Horocyclic Minkowski总和类似地定义。 We prove that when $A$ and $B$ are Borel-measurable, $$ \sqrt{ Area( [A:B]_λ )} \geq (1-λ) \cdot \sqrt{ Area(A) } + λ\cdot \sqrt{ Area(B) }, $$ where $Area$ stands for hyperbolic area, with equality when $A$ and $B$是双曲机平面中的同心盘。我们还证明了Prékopa-Leindler和Borell-Brascamp-Lieb的不平等现象。这些不等式略微偏离了曲率和布鲁恩·米科夫斯基类型不等式的度量测量空间范式，其中公制空间的结构施加在歧管上，相关的曲线必然是通过arclength参数参数的大地测量。
Given two non-empty subsets $A$ and $B$ of the hyperbolic plane $\mathbb{H}^2$, we define their horocyclic Minkowski sum with parameter $λ=1/2$ as the set $[A:B]_{1/2} \subseteq \mathbb{H}^2$ of all midpoints of horocycle curves connecting a point in $A$ with a point in $B$. These horocycle curves are parameterized by hyperbolic arclength, and the horocyclic Minkowski sum with parameter $0 < λ<1$ is defined analogously. We prove that when $A$ and $B$ are Borel-measurable, $$ \sqrt{ Area( [A:B]_λ )} \geq (1-λ) \cdot \sqrt{ Area(A) } + λ\cdot \sqrt{ Area(B) }, $$ where $Area$ stands for hyperbolic area, with equality when $A$ and $B$ are concentric discs in the hyperbolic plane. We also prove horocyclic versions of the Prékopa-Leindler and Borell-Brascamp-Lieb inequalities. These inequalities slightly deviate from the metric measure space paradigm on curvature and Brunn-Minkowski type inequalities, where the structure of a metric space is imposed on the manifold, and the relevant curves are necessarily geodesics parameterized by arclength.
