Separation-Sensitive Collision Detection for Convex Objects

We develop a class of new kinetic data structures for collision detection between moving convex polytopes; the performance of these structures is sensitive to the separation of the polytopes during their motion. For two convex polygons in the plane, let $D$ be the maximum diameter of the polygons, and let $s$ be the minimum distance between them during their motion. Our separation certificate changes $O(\log(D/s))$ times when the relative motion of the two polygons is a translation along a straight line or convex curve, $O(\sqrt{D/s})$ for translation along an algebraic trajectory, and $O(D/s)$ for algebraic rigid motion (translation and rotation). Each certificate update is performed in $O(\log(D/s))$ time. Variants of these data structures are also shown that exhibit \emph{hysteresis}---after a separation certificate fails, the new certificate cannot fail again until the objects have moved by some constant fraction of their current separation. We can then bound the number of events by the combinatorial size of a certain cover of the motion path by balls.Comment: 10 pages, 8 figures; to appear in Proc. 10th Annual ACM-SIAM Symposium on Discrete Algorithms, 1999; see also http://www.uiuc.edu/ph/www/jeffe/pubs/kollide.html ; v2 replaces submission with camera-ready versio

Sharp L^1 Poincare inequalities correspond to optimal hypersurface cuts

Let $\Omega \subset \mathbb{R}^n$ be a convex. If $u: \Omega \rightarrow \mathbb{R}$ has mean 0, then we have the classical Poincar\'{e} inequality \|u \|_{L^p} \leq c_p \mbox{diam}(\Omega) \| \nabla u \|_{L^p} with sharp constants $c_2 = 1/\pi$ (Payne \& Weinberger, 1960) and $c_1 = 1/2$ (Acosta \& Duran, 2005) independent of the dimension. The sharp constants $c_p$ for $1 < p < 2$ have recently been found by Ferone, Nitsch \& Trombetti (2012). The purpose of this short paper is to prove a much stronger inequality in the endpoint $L^1$: we combine results of Cianchi and Kannan, Lov\'{a}sz \& Simonovits to show that $$\left\|u\right\|_{L^{1}(\Omega)} \leq \frac{2}{\log{2}} M_{}(\Omega) \left\|\nabla u\right\|_{L^{1}(\Omega)}$$ where $M_{}(\Omega)$ is the average distance between a point in $\Omega$ and the center of gravity of $\Omega$. If $\Omega$ is a simplex, this yields an improvement by a factor of $\sim \sqrt{n}$ in $n$ dimensions. By interpolation, this implies that that for every convex $\Omega \subset \mathbb{R}^n$ and every $u:\Omega \rightarrow \mathbb{R}$ with mean 0 \left\|u\right\|_{L^{p}(\Omega)}\leq \left(\frac{2}{\log{2}} M_{}(\Omega) \right)^{\frac{1}{p}}\mbox{diam}(\Omega)^{1-\frac{1}{p}}\left\|\nabla u\right\|_{L^{p}(\Omega)}. Comment: New version with extension to L^p for p > 1, to be published in Archiv der Mathemati

Regular homotopy and total curvature

We consider properties of the total absolute geodesic curvature functional on circle immersions into a Riemann surface. In particular, we study its behavior under regular homotopies, its infima in regular homotopy classes, and the homotopy types of spaces of its local minima. We consider properties of the total curvature functional on the space of 2-sphere immersions into 3-space. We show that the infimum over all sphere eversions of the maximum of the total curvature during an eversion is at most 8\pi and we establish a non-injectivity result for local minima.Comment: This is the version published by Algebraic & Geometric Topology on 23 March 2006. arXiv admin note: this version concatenates two articles published in Algebraic & Geometric Topolog