From the pentagon onwards, the area of the regular convex polygon with n sides and unit diameter is greater for each odd number n than for the next even number n + 1. Moreover, from the heptagon onwards, the difference in areas decreases when n increases. Similar properties hold for the perimeter. A new proof of a result of Reinhardt follows.

We give a generalization of Bender's area-perimeter relation for plane lattice-point-free convex regions to simply connected regions, thus we solve a problem posed by M. Silver 10]. Further the result is used for a lattice version of the Dido problem.

In this paper, we present new algorithms for computing maximum perimeter and maximum area of the convex hull of imprecise inputs based on the parallel line segment model. The running time of our algorithms for maximum perimeter problem is O(n) which improve over the previous results of O(n) in [12]. For maximum are problem with different size of parallel line segments [12] gives O(n) algorithms...

Let S be a set of n points in the plane. We present data structures that solve range-aggregate query problems on three geometric extent measure problems. Using these data structures, we can report, for any axis-parallel query rectangle Q, the area/perimeter of the convex hull, the width, and the radius of the smallest enclosing disk of the points in S ∩Q.

We investigate the geometric properties of the convex hull over n successive positions of a planar random walk, with a symmetric continuous jump distribution. We derive the large n asymptotic behavior of the mean perimeter. In addition, we compute the mean area for the particular case of isotropic Gaussian jumps. While the leading terms of these asymptotics are universal, the subleading (correc...

We consider the problem of removing c points from a set S of n points so that the resulting point set has the smallest possible convex hull. Our main result is an O ( n (( 4c 2c ) (3c) + log n )) time algorithm that solves this problem when “smallest” is taken to mean least area or least perimeter.

We compare the perimeter measure with the Airault-Malliavin surface measure and we prove that all open convex subsets of abstract Wiener spaces have finite perimeter. By an explicit counter–example, we show that in general this is not true for compact convex domains.