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A spherical convex body R is called reduced if any convex body properly contained in R has thickness smaller than the thickness Δ (R) of R. We show that for every reduced spherical convex polygon V there exists its boundary point such that the disk of radius Δ (V) centered at this point contains V. [ABSTRACT FROM AUTHOR]

The univalent function ϕ SG (z) : = 2 / (1 + e − z) maps the open unit disk | z | < 1 onto a sigmoid domain | log (w / (2 − w)) | < 1. The sharp radii constants have been computed for the classes involving ratios f / g and g / (z p) of analytic functions f , g and p defined in | z | < 1 such that p is subordinate to ϕ SG , the other two ratios are subordinate to 1 + z , e z or z + 1 + z 2 and the quantity z f ′ / f lies in a certain region of the right half plane. [ABSTRACT FROM AUTHOR]

Similarly to the classic notion in Euclidean space, we call a set on the sphere S d complete, provided adding any extra point increases its diameter. Complete sets are convex bodies on S d . Our main theorem says that on S d complete bodies of diameter δ coincide with bodies of constant width δ . [ABSTRACT FROM AUTHOR]

After a few claims about lunes and convex sets on the d-dimensional sphere S d we present some relationships between the diameter, width and thickness of reduced convex bodies and bodies of constant diameter on S d . These relationships are applied for proving the final theorem, which permits to recognize if a Wulff shape in the Euclidean space is self-dual. [ABSTRACT FROM AUTHOR]

We prove that a smooth convex body of diameter δ < π 2 on the d-dimensional unit sphere S d is of constant diameter δ if and only if it is of constant width δ . We also show this equivalence for all convex bodies on S 2 . Since, as shown earlier, the equivalence on S d is true for every δ ≥ π 2 , the question whether spherical bodies of constant diameter and constant width on S d coincide remains open for non-smooth bodies on S d , where d ≥ 3 . [ABSTRACT FROM AUTHOR]

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The intersection L of two different non-opposite hemispheres G and H of the d-dimensional unit sphere Sd is called a lune. By the thickness of L we mean the distance of the centers of the (d-1)-dimensional hemispheres bounding L. For a hemisphere G supporting a convex body C⊂Sd we define widthG(C) as the thickness of the narrowest lune or lunes of the form G∩H containing C. If widthG(C)=w for every hemisphere G supporting C, we say that C is a body of constant width w. We present properties of these bodies. In particular, we prove that the diameter of any spherical body C of constant width w on Sd is w, and that if w<π2, then C is strictly convex. Moreover, we check when spherical bodies of constant width and constant diameter coincide. [ABSTRACT FROM AUTHOR]

Several subclasses of starlike functions are associated with regions in the right half plane of the complex plane, like half-plane, disks, sectors, parabolas and lemniscate of Bernoulli. For a normalized analytic function defined on the open unit disk belonging to certain well-known classes of functions associated with the above regions, we investigate the radius such that, for the function , lies in the lune defined by for all . [ABSTRACT FROM AUTHOR]

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For every hemisphere K supporting a convex body C on the sphere S we define the width of C determined by K. We show that it is a continuous function of the position of K. We prove that the diameter of every convex body $${C \subset S^d}$$ equals the maximum of the widths of C provided the diameter of C is at most $${\frac{\pi}{2}}$$ . In a natural way, we define spherical bodies of constant width. We also consider the thickness Δ( C) of C, i.e., the minimum width of C. A convex body $${R \subset S^d}$$ is said to be reduced if Δ( Z) < Δ( R) for every convex body Z properly contained in R. For instance, bodies of constant width on S and regular spherical odd-gons of thickness at most $${\frac{\pi}{2}}$$ on S are reduced. We prove that every reduced smooth spherical convex body is of constant width. [ABSTRACT FROM AUTHOR]

The Schwarz problem is explicitly solved for the inhomogeneous Cauchy–Riemann equation in a particular circular lens and two related complementary lunes. The solutions are given by the same analytical formula restricted to the respective domain. The parqueting-reflection method is used to construct the Cauchy–Schwarz representation formula leading to the Schwarz and thus the Poisson kernels for the three domains. [ABSTRACT FROM PUBLISHER]

Copyright of Cartographica is the property of University of Toronto Press and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)