The paper answers the three distinct questions of maximizing the perimeter, diameter and area of equilateral unit-width convex polygons. The solution to each of these problems is trivially unbounded when the number of sides is even. We show that when this number is odd, the optimal solution to these three problems is identical, and arbitrarily close to a trapezoid. The paper also considers the ...

An approach is presented for the enumeration of directed-convex polyominos that are not parallelogram polyominoes and we establish that there are ( 2n n−2 ) with a perimeter of 2n + 4. Finally using known results we prove that there are ( 2n n ) directed-convex polyominos with a perimeter of 2n + 4.

We compute exactly the mean perimeter and area of convex hull a $2$-d Brownian motion duration $t$ diffusion constant $D$, in presence resetting to origin at rate $r$. show that for any $t$, is given by $\langle L(t)\rangle= 2 \pi \sqrt{\frac{D}{r}}\, f_1(rt)$ A(t) \rangle= 2\pi\frac{D}{r}\, f_2(rt)$ where scaling functions $f_1(z)$ $f_2(z)$ are computed explicitly. For large $t\gg 1/r$, grows ...

Let S be a set of n points in the plane. We study the following problem: Partition S by a line into two subsets Sa and Sb such that maxff(Sa); f(Sb)g is minimal, where f is any monotone function de ned over 2 . We rst present a solution to the case where the points in S are the vertices of some convex polygon and apply it to some common cases | f(S0) is the perimeter, area, or width of the conv...

Wheat (Triticum aestivum L.) is known to be the world-leading cereal grain and the most important food in the world of agriculture.  Wheat offers a great wealth of material for genetic studies due to its wide ecological distribution and host of variation for various morphological and physiological characters.  To evaluate the genetic control of physical traits of grain in two crosses of winter ...