For two convex discs $K$ and $L$, we say that $K$ is $L$-convex if it is equal to the intersection of all translates of $L$ that contain $K$. In $L$-convexity the set $L$ plays a similar role as closed half-spaces do in the classical notion of convexity. We study the following probability model: Let $K$ and $L$ be $C^2_+$ smooth convex discs such that $K$ is $L$-convex. Select $n$ i.i.d. uniform random points $x_1,\ldots, x_n$ from $K$, and consider the intersection $K_{(n)}$ of all translates of $L$ that contain all of $x_1,\ldots, x_n$. The set $K_{(n)}$ is a random $L$-convex polygon in $K$. We study the expectation of the number of vertices $f_0(K_{(n)})$ and the missed area $A(K\setminus K_{n})$ as $n$ tends to infinity. We consider two special cases of the model. In the first case we assume that the maximum of the curvature of the boundary of $L$ is strictly less than $1$ and the minimum of the curvature of $K$ is larger than $1$. In this setting the expected number of vertices and missed area behave in a similar way as in the classical convex case and in the $r$-spindle convex case (when $L$ is a radius $r$ circular disc). The other case we study is when $K=L$. This setting is special in the sense that an interesting phenomenon occurs: the expected number of vertices tends to a finite limit depending only on $L$. This was previously observed in the special case when $L$ is a circle of radius $r$ (Fodor, Kevei and Vigh (2014)). We also determine the extrema of the limit of the expectation of the number of vertices of $L_{(n)}$ if $L$ is a convex discs of constant width $1$. The formulas we prove can be considered as generalizations of the corresponding $r$-spindle convex statements proved by Fodor, Kevei and Vigh (2014).

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关于广义圆盘多边形的随机近似

对于两个凸圆盘 $K$ 和 $L$，如果 $K$ 等于 $L$ 包含 $K$ 的所有平移的交集，我们说 $K$ 是 $L$-凸面。在$L$-convexity 中，集合$L$ 扮演的角色与封闭半空间在经典凸性概念中的作用相似。我们研究以下概率模型：设 $K$ 和 $L$ 是 $C^2_+$ 光滑凸盘，使得 $K$ 是 $L$-凸面。从 $K$ 中选择 $n$ iid 均匀随机点 $x_1,\ldots, x_n$，并考虑包含所有 $x_1,\ldots 的 $L$ 的所有平移的交集 $K_{(n)}$， x_n$。集合 $K_{(n)}$ 是 $K$ 中的随机 $L$-凸多边形。我们研究顶点数 $f_0(K_{(n)})$ 和遗漏区域 $A(K\setminus K_{n})$ 的期望值，因为 $n$ 趋于无穷大。我们考虑模型的两种特殊情况。在第一种情况下，我们假设 $L$ 边界的曲率最大值严格小于 $1$，而 $K$ 的曲率最小值大于 $1$。在这个设置中，顶点的预期数量和遗漏区域的行为方式与经典凸情况和 $r$-spindle 凸情况（当 $L$ 是半径 $r$ 圆盘时）的行为方式相似。我们研究的另一种情况是 $K=L$。这个设置是特殊的，因为一个有趣的现象发生了：顶点的预期数量趋向于一个仅取决于 $L$ 的有限限制。这是先前在 $L$ 是半径为 $r$ 的圆的特殊情况下观察到的（Fodor、Kevei 和 Vigh（2014））。如果 $L$ 是等宽 $1$ 的凸圆盘，我们还确定了 $L_{(n)}$ 顶点数的期望极限的极值。