A Generalization of the Convex Kakeya Problem

Given a set of line segments in the plane, not necessarily finite, what is a convex region of smallest area that contains a translate of each input segment? This question can be seen as a generalization of Kakeya's problem of finding a convex region of smallest area such that a needle can be rotated through 360 degrees within this region. We show that there is always an optimal region that is a triangle, and we give an optimal \Theta(n log n)-time algorithm to compute such a triangle for a given set of n segments. We also show that, if the goal is to minimize the perimeter of the region instead of its area, then placing the segments with their midpoint at the origin and taking their convex hull results in an optimal solution. Finally, we show that for any compact convex figure G, the smallest enclosing disk of G is a smallest-perimeter region containing a translate of every rotated copy of G.Comment: 14 pages, 9 figure

On the perimeters of simple polygons contained in a disk

A simple $n$-gon is a polygon with $n$ edges with each vertex belonging to exactly two edges and every other point belonging to at most one edge. Brass asked the following question: For $n \geq 5$ odd, what is the maximum perimeter of a simple $n$-gon contained in a Euclidean unit disk? In 2009, Audet, Hansen and Messine answered this question, and showed that the optimal configuration is an isosceles triangle with a multiple edge, inscribed in the disk. In this note we give a shorter and simpler proof of their result, which we generalize also for hyperbolic disks, and for spherical disks of sufficiently small radii.Comment: 6 pages, 2 figure

Compact convex sets of the plane and probability theory

The Gauss-Minkowski correspondence in $\mathbb{R}^2$ states the existence of a homeomorphism between the probability measures $\mu$ on $[0,2\pi]$ such that $\int_0^{2\pi} e^{ix}d\mu(x)=0$ and the compact convex sets (CCS) of the plane with perimeter~1. In this article, we bring out explicit formulas relating the border of a CCS to its probability measure. As a consequence, we show that some natural operations on CCS -- for example, the Minkowski sum -- have natural translations in terms of probability measure operations, and reciprocally, the convolution of measures translates into a new notion of convolution of CCS. Additionally, we give a proof that a polygonal curve associated with a sample of $n$ random variables (satisfying $\int_0^{2\pi} e^{ix}d\mu(x)=0$) converges to a CCS associated with $\mu$ at speed $\sqrt{n}$, a result much similar to the convergence of the empirical process in statistics. Finally, we employ this correspondence to present models of smooth random CCS and simulations

Morphological Image Analysis of Quantum Motion in Billiards

Morphological image analysis is applied to the time evolution of the probability distribution of a quantum particle moving in two and three-dimensional billiards. It is shown that the time-averaged Euler characteristic of the probability density provides a well defined quantity to distinguish between classically integrable and non-integrable billiards. In three dimensions the time-averaged mean breadth of the probability density may also be used for this purpose.Comment: Major revision. Changes include a more detailed discussion of the theory and results for 3 dimensions. Now: 10 pages, 9 figures (some are colored), 3 table

On the quantitative isoperimetric inequality in the plane with the barycentric distance

In this paper we study the following quantitative isoperimetric inequality in the plane: $\lambda_0^2(\Omega) \leq C \delta(\Omega)$ where $\delta$ is the isoperimetric deficit and $\lambda_0$ is the barycentric asymmetry. Our aim is to generalize some results obtained by B. Fuglede in \cite{Fu93Geometriae}. For that purpose, we consider the shape optimization problem: minimize the ratio $\delta(\Omega)/\lambda_0^2(\Omega)$ in the class of compact connected sets and in the class of convex sets

Neuromorphometric characterization with shape functionals

This work presents a procedure to extract morphological information from neuronal cells based on the variation of shape functionals as the cell geometry undergoes a dilation through a wide interval of spatial scales. The targeted shapes are alpha and beta cat retinal ganglion cells, which are characterized by different ranges of dendritic field diameter. Image functionals are expected to act as descriptors of the shape, gathering relevant geometric and topological features of the complex cell form. We present a comparative study of classification performance of additive shape descriptors, namely, Minkowski functionals, and the nonadditive multiscale fractal. We found that the proposed measures perform efficiently the task of identifying the two main classes alpha and beta based solely on scale invariant information, while also providing intraclass morphological assessment

Deformations of Toric Singularities and Fractional Branes

Fractional branes added to a large stack of D3-branes at the singularity of a Calabi-Yau cone modify the quiver gauge theory breaking conformal invariance and leading to different kinds of IR behaviors. For toric singularities admitting complex deformations we propose a simple method that allows to compute the anomaly free rank distributions in the gauge theory corresponding to the fractional deformation branes. This algorithm fits Altmann's rule of decomposition of the toric diagram into a Minkowski sum of polytopes. More generally we suggest how different IR behaviors triggered by fractional branes can be classified by looking at suitable weights associated with the external legs of the (p,q) web. We check the proposal on many examples and match in some interesting cases the moduli space of the gauge theory with the deformed geometry.Comment: 40 pages, 23 figures; typos correcte