Planar Projections and Second Intrinsic Volume

Consider random shadows of a cube and of a regular tetrahedron. Area and perimeter of the former are positively dependent (with correlation 0.915...), whereas area and perimeter of the latter appear to be negatively dependent. This is only one result of many, all involving generalizations of mean width.Comment: 13 page

CLT Variance Associated with Baxendale's SDE

Simple analysis of the leftmost eigenvalue of Ince's equation (a boundary value problem with periodicity) resolves an open issue surrounding a stochastic Lyapunov exponent. Numerical verification is also provided.Comment: 6 page

Triangles Formed via Poisson Nearest Neighbors

We start with certain joint densities (for sides and for angles) corresponding to pinned Poissonian triangles in the plane, then discuss analogous results for staked and anchored triangles.Comment: 16 pages, 4 figure

Mean Width of a Regular Simplex

The mean width is a measure on n-dimensional convex bodies. An integral formula for the mean width of a regular n-simplex appeared in the electrical engineering literature in 1997. As a consequence, expressions for the expected range of a sample of n+1 normally distributed variables, for n<=6, carry over to widths of regular n-simplices. As another consequence, precise asymptotics for the mean width become available as n->infty.Comment: 12 page

In Limbo: Three Triangle Centers

Yet more candidates are proposed for inclusion in the Encyclopedia of Triangle Centers. Our focus is entirely on simple calculations.Comment: 13 pages, 4 figures (correctly placed in v2

Covering a Sphere with Four Random Circular Caps

Let p(w) denote the probability that four random circular caps of angular radius 70deg<w<90deg cover the unit sphere S^2. An exact expression for p(w) is unknown. We give nontrivial lower bounds for p(w) when w>84deg; no improvement on the inequality p(w)>=0 for w<84deg is yet feasible. A dual problem involving randomly inscribed well-centered tetrahedra is also examined.Comment: 10 pages, 2 figure

Oblique Circular Cones and Cylinders

Surface area and mean width of a cylinder (the convex hull of two parallel disks) in R^3 are computed. It is more difficult to obtain analogous results for a cone (the convex hull of a disk D and a point p). Oblique formulas for mean width, as well as those for mean curvature, are new. Let L denote the unique diameter of D whose endpoints are equidistant from p. We conclude with a question involving the plane that bisects the cone and contains {p,L}, as p varies. What is the minimum ratio of the smaller measure to the larger?Comment: 16 page

Appell F1 and Conformal Mapping

This is the last of a trilogy of papers on triangle centers. A fairly obscure "conformal center of gravity" is computed for the class of all isosceles triangles. This calculation appears to be new. A byproduct is the logarithmic capacity or transfinite diameter of such, yielding results consistent with Haegi (1951).Comment: 12 pages, 3 figure

0-Pierced Triangles within a Poisson Overlay

Let the Euclidean plane be simultaneously and independently endowed with a Poisson point process and a Poisson line process, each of unit intensity. Consider a triangle T whose vertices all belong to the point process. The triangle is 0-pierced if no member of the line process intersects any side of T. Our starting point is Ambartzumian's 1982 joint density for angles of T; our exposition is elementary and raises several unanswered questions.Comment: 10 pages, 7 figure

Mean Width of a Regular Cross-Polytope

The expected range of a sample of n+1 normally distributed variables is known to be related to the mean width of a regular n-simplex. We show that the expected maximum mu_n of a sample of n half-normally distributed variables is related to the mean width of a regular n-crosspolytope. Both of these relations have mean square counterparts. An expression for mu_5 is found and is believed to be new.Comment: 11 page