Candidate Sets for Complex Interval

Uncertainty of measuring complex-valued physical quantities can be described by complex sets. These sets can have complicated shapes, so we would like to nd a good approximating family of sets. Which approximating family is the best? We reduce the corresponding optimization problem to a geometric one: namely, we prove that, under some reasonable conditions, an optimal family must be shift-, rotation-and scale-invariant. We then use this geometric reduction to conclude that the best approximating low-dimensional families consist of sets with linear or circular boundaries. This result is consistent with the fact that such sets have indeed been successful in computations. It stimulates to study further candidates. A practical problem leading to complex sets. Many physical quantities are complex-valued: wave function in quantum mechanics, complex amplitude and impedance in electrical engineering, etc. Due to measurement uncertainty, after measuring a value of a physical quantity, we do not get its exact value, we only get a set of possible values of this quantity. The shapes of these sets can be very complicated, so we would like to approximate them by simpler shapes from an approximating family. Which family should we choose? In 1-D case, a similar problem has a simple solution: we choose the family of all (real) intervals. This family has many good properties; in particular, it is closed under point-wise arithmetic operations A B = fa b j a 2 A; b 2 Bg such as addition, subtraction, and multiplication, which makes this family perfect for the analysis of how these measurement results get processed in a computer. Unfortunately, for complex sets, no nite-dimensional family containing real intervals is closed under these operations Nickel 1980]; moreover, no nite-dimensional family containing real intervals is closed under addition and under multiplication by complex numbers. This negative result has a clear geometric meaning, due to the fact that adding a complex number means a shift, and multiplication by a complex number exp(i) means rotation by an angle and scaling times. So, Nickel's negative result means it is impossible to have a nite-dimensional family of complex sets which would be closed under addition, invariant under shift, rotation, and scaling, and contain real intervals. Since we cannot have an approximating family which satisses all desired properties, we must therefore use families which satisfy only some of them. Several families have been proposed: boxes, polygons, circles, el-lipsoids, etc. Some families approximate better, some approximate worse. So, …