On the size of approximately convex sets in normed spaces

Let X be a normed space. A subset A of X is approximately convex if $d(ta+(1-t)b,A) \le 1$ for all $a,b \in A$ and $t \in [0,1]$ where $d(x,A)$ is the distance of $x$ to $A$. Let \Co(A) be the convex hull and \diam(A) the diameter of $A$. We prove that every $n$-dimensional normed space contains approximately convex sets $A$ with \mathcal{H}(A,\Co(A))\ge \log_2n-1 and \diam(A) \le C\sqrt n(\ln n)^2, where $\mathcal{H}$ denotes the Hausdorff distance. These estimates are reasonably sharp. For every $D>0$, we construct worst possible approximately convex sets in $C[0,1]$ such that \mathcal{H}(A,\Co(A))=\diam(A)=D. Several results pertaining to the Hyers-Ulam stability theorem are also proved.Comment: 32 pages. See also http://www.math.sc.edu/~howard

Extremal Approximately Convex Functions and Estimating the Size of Convex Hulls

A real valued function $f$ defined on a convex $K$ is anemconvex function iff it satisfies $$f((x+y)/2) \le (f(x)+f(y))/2 + 1.$$ A thorough study of approximately convex functions is made. The principal results are a sharp universal upper bound for lower semi-continuous approximately convex functions that vanish on the vertices of a simplex and an explicit description of the unique largest bounded approximately convex function~$E$ vanishing on the vertices of a simplex. A set $A$ in a normed space is an approximately convex set iff for all $a,b\in A$ the distance of the midpoint $(a+b)/2$ to $A$ is $\le 1$. The bounds on approximately convex functions are used to show that in $\R^n$ with the Euclidean norm, for any approximately convex set $A$, any point $z$ of the convex hull of $A$ is at a distance of at most $[\log_2(n-1)]+1+(n-1)/2^{[\log_2(n-1)]}$ from $A$. Examples are given to show this is the sharp bound. Bounds for general norms on $R^n$ are also given.Comment: 39 pages. See also http://www.math.sc.edu/~howard

A New Species of Draeculacephala from California (Homoptera, Cicadellidae)

Author Institution: Entomological Laboratories, Ohio State University and Division of Entomology, University of Californi

Interleukin-6, age, and corpus callosum integrity.

The contribution of inflammation to deleterious aging outcomes is increasingly recognized; however, little is known about the complex relationship between interleukin-6 (IL-6) and brain structure, or how this association might change with increasing age. We examined the association between IL-6, white matter integrity, and cognition in 151 community dwelling older adults, and tested whether age moderated these associations. Blood levels of IL-6 and vascular risk (e.g., homocysteine), as well as health history information, were collected. Processing speed assessments were administered to assess cognitive functioning, and we employed tract-based spatial statistics to examine whole brain white matter and regions of interest. Given the association between inflammation, vascular risk, and corpus callosum (CC) integrity, fractional anisotropy (FA) of the genu, body, and splenium represented our primary dependent variables. Whole brain analysis revealed an inverse association between IL-6 and CC fractional anisotropy. Subsequent ROI linear regression and ridge regression analyses indicated that the magnitude of this effect increased with age; thus, older individuals with higher IL-6 levels displayed lower white matter integrity. Finally, higher IL-6 levels were related to worse processing speed; this association was moderated by age, and was not fully accounted for by CC volume. This study highlights that at older ages, the association between higher IL-6 levels and lower white matter integrity is more pronounced; furthermore, it underscores the important, albeit burgeoning role of inflammatory processes in cognitive aging trajectories

Copyright & Privacy - Through the Political Lens, 4 J. Marshall Rev. Intell. Prop. L. 306 (2005)

Veteran beltway players discuss the politics of P2P technology and Privacy. How far can or should Congress go? Can the United States export its values or its laws in this area? Are content owners in a losing Luddite struggle? What is the role of litigators, lobbyists and legislators in this war