arXiv subjects
Publications and source records attributed to Viktor Vígh.
We prove asymptotic lower bounds on the variance of the number of vertices and missed area of random disc-polygons in convex discs whose boundary is $C_+^2$ smooth. The established lower bounds are of the same order as the upper bounds proved previously by Fodor and V\'ıgh (2018).
We prove asymptotic formulas for the expectation of the vertex number and missed area of uniform random disc-polygons in convex disc-polygons. Our statements are the $r$-convex analogues of the classical results of Rényi and Sulanke (1964) about random polygons in convex polygons.
In this note we prove that any monohedral tiling of the closed circular unit disc with $k \leq 3$ topological discs as tiles has a $k$-fold rotational symmetry. This result yields the first nontrivial estimate about the minimum number of tiles in a monohedral tiling of the circular disc in which not all tiles contain the center, and the first step towards answering a question of Stein appearing in the problem book of Croft, Falconer and Guy in 1994.
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 Vígh (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 Vígh (2014).
In this paper we prove asymptotic upper bounds on the variance of the number of vertices and missed area of inscribed random disc-polygons in smooth convex discs whose boundary is $C^2_+$. We also consider a circumscribed variant of this probability model in which the convex disc is approximated by the intersection of random circles.
Let $K$ and $K_0$ be convex bodies in $\mathbb{R}^d$, such that $K$ contains the origin, and define the process $(K_n, p_n)$, $n \geq 0$, as follows: let $p_{n+1}$ be a uniform random point in $K_n$, and set $K_{n+1} = K_n \cap (p_{n+1} + K)$. Clearly, $(K_n)$ is a nested sequence of convex bodies which converge to a non-empty limit object, again a convex body in $\mathbb{R}^d$. We study this process for $K$ being a regular simplex, a cube, or a regular convex polygon with an odd number of vertices. We also derive some new results in one dimension for non-uniform distributions.
Let S(1) be the segment [-1,1], and define the segments S(n) recursively in the following manner: let S(n+1) be the intersection of S(n) and a(n+1) + S(1), where the point a(n+1) is chosen randomly on the segment S(n) with uniform distribution. For the radius r(n) of S(n) we prove that n(r(n) - 1//2) converges in distribution to an exponential law, and we also show that the centre of the limiting unit interval has arcsine distribution.
Let $K$ be a $d$ dimensional convex body with a twice continuously differentiable boundary and everywhere positive Gauss-Kronecker curvature. Denote by $K_n$ the convex hull of $n$ points chosen randomly and independently from $K$ according to the uniform distribution. Matching lower and upper bounds are obtained for the orders of magnitude of the variances of the $s$-th intrinsic volumes $V_s(K_n)$ of $K_n$ for $s\in\{1, ..., d\}$. Furthermore, strong laws of large numbers are proved for the intrinsic volumes of $K_n$. The essential tools are the Economic Cap Covering Theorem of Bárány and Larman, and the Efron-Stein jackknife inequality.