Geominimal surface area

Geometriae Dedicata -     

On random surface area

Consider a random smooth Gaussian field G(x): F ? \mathbbR F \to \mathbb{R} , where F is a compact set in \mathbbR^d {\mathbb{R}^d} . We derive a formula for the average area of a surface determined by the equation G(x) = 0 and give some applications. As an auxiliary result, we obtain an integral expression for the area of a surface determined by zeros of a nonrandom smooth field. Bibliography: 13 titles.     

Gibbs' phenomenon and surface area

If a function is of bounded variation on and is a positive approximate identity, we prove that the area of the graph of converges from below to the relaxed area of the graph of . Moreover we give asymptotic estimates for the area of the graph of the square partial sums of multiple Fourier series of functions with suitable discontinuities.

     

Concentration of area in half-planes

For the standard class of normalized univalent functions analytic in the unit disk , we consider a problem on the minimal area of the image concentrated in any given half-plane. This question is related to a well-known problem posed by A. W. Goodman in 1949 that regards minimizing area covered by analytic univalent functions under certain geometric constraints. An interesting aspect of this problem is the unexpected behavior of the candidates for extremal functions constructed via geometric considerations.

     

The circumpolygon with maximal area

The convex circumpolygon with maximal area to a given convex polygon can be determined by means of dynamic programming. The effort of this method increases cubically with respect to the number of sides. It is further shown that the optimal circum-polygon can be constructed with ruler and circle. The applied version of dynamic programming can be also used for solving Steiner's problem of the inpoiygon with minimal circumference but it demands a higher effort than Phú's method.     

Contributions to affine surface area

Representations of equiaffine surface area, due to Leichtweiß resp. Schütt &; Werner, are generalized top-affine surface area measures. We provide a direct proof which shows that these representations coincide. In addition, we establish two theoremes which in particular characterize all those convex bodies geometrically for which the affine surface area is positive. The present approach also leads to proofs of the equiaffine isoperimetric inequality and the Blaschke-Santaló inequality, including the characterization of the case of equality.     

The area within a curve

The area of a simple closed convex curve can be estimated in terms of the number of points of a square lattice that lie within the curve. We obtain the usual error bound without integration using a form of the Hardy—Littlewood—Ramanujan circle method, and also present simple estimates for the mean square error.     

Area and the area formula

Conferenza tenuta da Mariano Giaquinta il 28 settembre 1992     

BMO-boundedness of Lusin's area function

We prove boundedness of the Lusin's area operatorS _T on the space BMO(R ⁿ ).     

Nonsymmetric systems and area integral estimates

Even though the Dirichlet problem on Lipschitz domains is not always solvable for nonsymmetric strongly elliptic systems, so that many results and techniques from the symmetric systems are unavailable, there are some similarities with the symmetric systems. We show that the nontangential maximal function and the square function of a solution are equivalent and that there is a Fatou theorem for these solutions.

     

Filling an area discretely and continuously

The literature dealing with student understanding of integration in general and the Fundamental Theorem of Calculus in particular suggests that although students can integrate properly, they understand little about the process that leads to the definite integral. The definite integral is naturally connected to the antiderivative, the area under the curve and the limit of Riemann sums; these three conceptualizations of the definite integral are useful in different contexts and provide students with what it takes to interpret the definite integrals. Research shows that students rarely invoke the multiplicatively-based summation conception of the definite integral although it is essential for evaluating line integrals, surface integrals and volumes. This paper describes a teaching module that promotes understanding as well as activating all three conceptualizations of the definite integral through motivating the accumulation area function and the results in the Fundamental Theorems of Calculus.     

Approximation in area of continuous graphs

Summary We prove that graphs of continuous maps with finite area can be approximated weakly as currents and in area by graphs of smooth maps.This article was processed by the author using the style filepljour1m from Springer-Verlag.     

Mappings preserving area 1 of triangles

This paper is concerned with the characterization of area-preserving mappings of real inner product spaces.