In this note we present a reverse isoperimetric inequality for closed convex curves, which states that if γ is a closed strictly convex plane curve with length L and enclosing an area A, then one gets L ≤ 4π(A+ |Ã|), where à denotes the oriented area of the domain enclosed by the locus of curvature centers of γ, and the equality holds if and only if γ is a circle. MSC 2000: 52A38, 52A40

Let K be a convex body in R and B be the Euclidean unit ball in R. We show that limt→0 |K| − |Kt| |B| − |Bt| = as(K) as(B) , where as(K) respectively as(B) is the affine surface area of K respectively B and {Kt}t≥0, {Bt}t≥0 are general families of convex bodies constructed from K, B satifying certain conditions. As a corollary we get results obtained in [M-W], [Schm],[S-W] and[W]. The affine su...

Curvature measure is one of the basic notion in the theory of convex bodies. Together with surface area measures, they play fundamental roles in the study of convex bodies. They are closely related to the differential geometry and integral geometry of convex hypersurfaces. Let Ω is a bounded convex body in R with C2 boundary M , the corresponding curvature measures and surface area measures of ...

The dot comparison task, in which participants select the more numerous of two dot arrays, has become the predominant method of assessing Approximate Number System (ANS) acuity. Creation of the dot arrays requires the manipulation of visual characteristics, such as dot size and convex hull. For the task to provide a valid measure of ANS acuity, participants must ignore these characteristics and...

We consider the problem of finding the maximum area parallelogram (MAP) inside a given convex polygon. Our main result is an algorithm for computing the MAP in an n-sided polygon in O(n) time. Achieving this running time requires proving several new structural properties of the MAP. Our algorithm actually computes all the locally maximal area parallelograms (LMAPs). In addition to the algorithm...

in this paper we introduce a sequential block iterative method and its simultaneous version with op-timal combination of weights (instead of convex combination) for solving convex feasibility problems.when the intersection of the given family of convex sets is nonempty, it is shown that any sequencegenerated by the given algorithms converges to a feasible point. additionally for linear feasibil...

We present a new polygon decomposition problem, the anchored area partition problem, which has applications to a multiple-robot terrain-covering problem. This problem concerns dividing a given polygon P into n polygonal pieces, each of a speciied area and each containing a certain point (site) on its boundary. We rst present the algorithm for the case when P is convex and contains no holes. The...

Denote by Km the mirror image of a planar convex body K in a straight line m. It is easy to show that K∗ m = conv(K ∪ Km) is the smallest (by inclusion) convex body whose axis of symmetry is m and which contains K. The ratio axs(K) of the area of K to the minimum area of K∗ m is a measure of axial symmetry of K. A question is how to find a line m in order to guarantee that K∗ m be of the smalle...