carpal tunnel syndrome (cts) is the most common entrapment neuropathy. it accounts 90% of all entrapment neuropathies all over the world. ultrasound is a non-invasive, cost effective and available para-clinical method which could be applied for cts diagnosis. cross-sectional area of the median nerve at the level of the inlet is considered as a diagnostic criterion in cts cases. in this study, t...

We consider the problem of minimizing a functional (like the area, perimeter, surface) within the class of convex bodies whose support functions are trigonometric polynomials. The convexity constraint is transformed via the Fejér-Riesz theorem on positive trigonometric polynomials into a semidefinite programming problem. Several problems such as the minimization of the area in the class of cons...

In this paper we study the following problem: how to divide a cake among the children attending a birthday party such that all the children get the same amount of cake and the same amount of icing. This leads us to the study of the following. A perfect k-partitioning of a convex set S is a partitioning of S into k convex pieces such that each piece has the same area and 1 k of the perimeter of ...

In this paper we summarize the known results and the main tools concerning complete systems of inequalities for families of convex sets. We discuss also the possibility of using these systems to determine particular subfamilies of planar convex sets with specific geometric significance. We also analyze complete systems of inequalities for 3-rotationally symmetric planar convex sets concerning t...

In this paper we study the following problem: how to divide a cake among the children attending a birthday party such that all the children get the same amount of cake and the same amount of icing. This leads us to the study of the following. A perfect k-partitioning of a convex set S is a partitioning of S into k convex pieces such that each piece has the same area and 1 k of the perimeter of ...

We solve a problem of R. Nandakumar by proving that there is no tiling of the plane with pairwise noncongruent triangles of equal area and equal perimeter. We also show that no convex polygon with more than three sides can be tiled with finitely many triangles such that no pair of them share a full side.

Let S be a set of n > 2 points in the plane whose convex hull has perimeter t. Given a number P ≥ t, we study the following problem: Of all curves of perimeter P that enclose S, which is the curve that encloses the maximum area? In this paper, we give a complete characterization of this curve. We show that there are cases where this curve cannot be computed exactly and provide an O(n logn)-time...

A four part dissection and rearrangement provides a new proof of the isoperimetric inequality in the plane as well as a new approach to Bonnesen-type error estimates for the isoperimetric deficit of compact convex sets and of star bodies that are centrally symmetric with respect to the origin. An isoperimetric inequality in R bounds the area (or related functional) of a compact set by some func...