Slices, multiplicity, and Lebesgue area

The Additivity of the Lebesgue Area

A triple of continuous functions T: x(ut u ), i = l, 2, 3, defined on a closed square Q [ O g w x ^ l , 0 ^ w 2 ^ l ] represents a surface © (1.6, 1.17, 1.21). If r is any closed rectangle in Q then we may speak of the triple Tr consisting of the above triple T with its range of definition restricted to r. This triple generates a surface @(r). If r\ and r2 have no interior points in common, and...

The Isoperimetric Inequality and the Lebesgue Definition of Surface Area

The literature of these classical isoperimetric inequalities is very extensive (comprehensive presentations may be found in Blaschke [l] and Bonnesen [l](1)). In most instances, only convex curves and convex surfaces are considered, or else it is assumed that the curves and surfaces involved are sufficiently regular to permit the use of the classical formulas for the quantities a(C), 1(C), A(S)...

Topological Methods in the Theory of Lebesgue Area

Trees, slices, and wheels: On the floorplan area minimization problem

Hierarchical approaches to floorplan design usually use trees to represent hierarchical floorplans, defining topological relations between a set of components to be placed on a (circuit) board. This placement is then optimized to minimize chip area, perimeter, or other such cost measurements. This work places the problem of minimizing hierarchical floorplans of order k within the appropriate th...

Lebesgue functions and Lebesgue constants in polynomial interpolation

The Lebesgue constant is a valuable numerical instrument for linear interpolation because it provides a measure of how close the interpolant of a function is to the best polynomial approximant of the function. Moreover, if the interpolant is computed by using the Lagrange basis, then the Lebesgue constant also expresses the conditioning of the interpolation problem. In addition, many publicatio...