We consider the following geometric optimization problem: find a maximum-area rectangle and a maximum-perimeter rectangle contained in a given convex polygon with n vertices. We give exact algorithms that solve these problems in time O(n). We also give (1 − ε)-approximation algorithms that take time O(ε−3/2 + ε−1/2 log n) for maximizing the area and O(ε−3 + ε−1 log n) for maximizing the perimeter.

Given a set P = {P0, . . . , Pk−1} of k convex polygons having n vertices in total in the plane, we consider the problem of finding k translations τ0, . . . , τk−1 of P0, . . . , Pk−1 such that the translated copies τiPi are pairwise disjoint and the area or the perimeter of the convex hull of ⋃k−1 i=0 τiPi is minimized. When k = 2, the problem can be solved in linear time but no previous work ...

We study fully convex polygons with a given area, and variable perimeter length on square and hexagonal lattices. We attach a weight tm to a convex polygon of perimeter m and show that the sum of weights of all polygons with a fixed area s varies as s(-theta(conv))eK(t)square root(s) for large s and t less than a critical threshold tc, where K(t) is a t-dependent constant, and theta(conv) is a ...

Given a set P of n points in the plane and a number k, we want to find a polygon ~ with vertices in P of minimum area that satisfies one of the following properties: (1) cK is a convex k-gon, (2) ~ is an empty convex k-gon, or (3) ~ is the convex hull of exactly k points of P. We give algorithms for solving each of these three problems in time O(kn3). The space complexity is O(n) for k = 4 and ...

A hyperconvex disc of radius r is a planar set with nonempty interior that is the intersection of closed circular discs of radius r. A convex disc-polygon of radius r is a set with nonempty interior that is the intersection of a finite number of closed circular discs of radius r. We prove that the maximum area and perimeter of convex disc-n-gons of radius r contained in a hyperconvex disc of ra...

We prove large deviations principles (LDPs) for the perimeter and area of convex hull a planar random walk with finite Laplace transform its increments.

In this paper a Blaschke-Santaló diagram involving the area, the perimeter and the elastic energy of planar convex bodies is considered. More precisely we give a description of set