Relative convex hulls have been suggested for multigridconvergent surface area estimation. Besides the existence of a convergence theorem there is no efficient algorithmic solution so far for calculating 3D relative convex hulls. This article discusses an approximative solution based on minimum-length polygon calculations. It is illustrated that this approximative calculation also proves (exper...

A half-plane probe through a polygon measures the area of intersection between a half-plane and the polygon. We dev elop techniques based on xray probing to determine convex n-gons in 7n + 7 half-plane probes. We also show n + 1 half-plane probes are sufficient to verify a specified convex polygon and prove linear lower bounds for determination and verification.

We show that every rigid motion invariant and upper semicontinuous valuation on the space of convex discs is a linear combination of the Euler characteristic, the length, the area, and a suitable curvature integral of the convex disc. 1991 AMS subject classification: 52A10, 53A04

In this paper we consider convex improper affine maps of the 3-dimensional affine space and classify their singularities. The main tool developed is a generating family with properties that closely resembles the area function for non-convex improper affine maps. Mathematics Subject Classification (2010). 53A15.

Let n ≥ 3 be an integer. A convex lattice n-gon is a polygon whose n vertices are points on the integer lattice Z 2 and whose interior angles are strictly less than π. Let a n denote the least possible area enclosed by a convex lattice n-gon, then [1, 2, 3] {a n } ∞ n=3 = n 1 2

‎We use of two notions functionally convex (briefly‎, ‎F--convex) and functionally closed (briefly‎, ‎F--closed) in functional analysis and obtain more results‎. ‎We show that if $lbrace A_{alpha} rbrace _{alpha in I}$ is a family $F$--convex subsets with non empty intersection of a Banach space $X$‎, ‎then $bigcup_{alphain I}A_{alpha}$ is F--convex‎. ‎Moreover‎, ‎we introduce new definition o...