arXiv subjects
Publications and source records attributed to Daniel A. Klain.
Two people meet in a coffeehouse and decide to share one dessert from a menu of several possible choices. How should they choose which one? A method is presented that is intended to be practical, avoiding the need for long negotiations or complete menu rankings, while offering a high likelihood of satisfaction and fairness to both participants. The method is also extended to parties of 3 or more people sharing one item from a menu of several choices.
If the four triangular facets of a tetrahedron can be partitioned into pairs having the same area, then the triangles in each pair must be congruent to one another. A Heron-style formula is then derived for the volume of a tetrahedron having this kind of symmetry.
This note offers a probabilistic proof of Girard's angle excess formula for the area of a spherical triangle, based on the observation that an unbounded 3-dimensional convex cone, with single vertex at the origin, has only three kinds of 2-dimensional orthogonal projections: a 2-dimensional convex cone with one vertex, a 2-dimensional half-plane (this is an outcome of probability zero), and a 2-dimensional plane.
Let $v_1, ..., v_m$ be a finite set of unit vectors in $\RR^n$. Suppose that an infinite sequence of Steiner symmetrizations are applied to a compact convex set $K$ in $\RR^n$, where each of the symmetrizations is taken with respect to a direction from among the $v_i$. Then the resulting sequence of Steiner symmetrals always converges, and the limiting body is symmetric under reflection in any of the directions $v_i$ that appear infinitely often in the sequence. In particular, an infinite periodic sequence of Steiner symmetrizations always converges, and the set functional determined by this infinite process is always idempotent.
For n >= 2 a construction is given for a large family of compact convex sets K and L in n-dimensional Euclidean space such that the orthogonal projection L_u onto the subspace u^\perp contains a translate of the corresponding projection K_u for every direction u, while the volumes of K and L satisfy V_n(K) > V_n(L). It is subsequently shown that, if the orthogonal projection L_u onto the subspace u^\perp contains a translate of K_u for every direction u, then the set (n/(n-1))L contains a translate of K. If follows that V_n(K) <= (n/(n-1))^n V_n(L). In particular, we derive a universal constant bound V_n(K) <= 2.942 V_n(L), independent of the dimension n of the ambient space. Related results are obtained for projections onto subspaces of some fixed intermediate co-dimension. Open questions and conjectures are also posed.
A countable dense set of directions is sufficient for Steiner symmetrization, but the order of directions matters.
Let K and L be compact convex sets in R^n. The following two statements are shown to be equivalent: (i) For every polytope Q inside K having at most n+1 vertices, L contains a translate of Q. (ii) L contains a translate of K. Let 1 <= d <= n-1. It is also shown that the following two statements are equivalent: (i) For every polytope Q inside K having at most d+1 vertices, L contains a translate of Q. (ii) For every d-dimensional subspace W, the orthogonal projection of the set L onto W contains a translate of the corresponding projection of the set K onto W. It is then shown that, if K is a compact convex set in R^n having at least d+2 exposed points, then there exists a compact convex set L such that every d-dimensional orthogonal projection of L contains a translate of the corresponding projection of K, while L does not contain a translate of K. In particular, such a convex body L exists whenever dim(K) > d.
This article describes a new proof of the equality condition for the Brunn-Minkowski inequality.
Let L be a compact convex set in R^n, and let 1 <= d <= n-1. The set L is defined to be d-decomposable if L is a direct Minkowski sum (affine Cartesian product) of two or more convex bodies each of dimension at most d. A compact convex set L is called d-reliable if, whenever each d-dimensional orthogonal projection of L contains a translate of the corresponding d-dimensional projection of a compact convex set K, it must follow that L contains a translate of K. It is shown that, for 1 <= d <= n-1: (1) d-decomposability implies d-reliability. (2) A compact convex set L in R^n is d-reliable if and only if, for all m >= d+2, no m unit normals to regular boundary points of L form the outer unit normals of a (m-1)-dimensional simplex. (3) Smooth convex bodies are not d-reliable. (4) A compact convex set L in R^n is 1-reliable if and only if L is 1-decomposable (i.e. a parallelotope). (5) A centrally symmetric compact convex set L in R^n is 2-reliable if and only if L is 2-decomposable. However, there are non-centered 2-reliable convex bodies that are not 2-decomposable.
For each i = 1, ..., n constructions are given for convex bodies K and L in n-dimensional Euclidean space such that each rank i orthogonal projection of K can be translated inside the corresponding projection of L, even though K has strictly larger m-th intrinsic volumes (i.e. V_m(K) > V_m(L)) for all m > i. It is then shown that, for each i = 1, ..., n, there is a class of bodies C{n,i}, called i-cylinder bodies of R^n, such that, if the body L with i-dimensional covering shadows is an i-cylinder body, then K will have smaller n-volume than L. The families C{n,i} are shown to form a strictly increasing chain of subsets C{n,1} < C{n,2} < ... < C{n,n-1} < C{n,n}, where C{n,1} is precisely the collection of centrally symmetric compact convex sets in n-dimensional space, while C{n,n} is the collection of all compact convex sets in n-dimensional space. Members of each family C{n,i} are seen to play a fundamental role in relating covering conditions for projections to the theory of mixed volumes, and members of C{n,i} are shown to satisfy certain geometric inequalities. Related open questions are also posed.
We use a probabilistic interpretation of solid angles to generalize the well-known fact that the inner angles of a triangle sum to 180 degrees. For the 3-dimensional case, we show that the sum of the solid inner vertex angles of a tetrahedron T, divided by 2*pi, gives the probability that an orthogonal projection of T onto a random 2-plane is a triangle. More generally, it is shown that the sum of the (solid) inner vertex angles of an n-simplex S, normalized by the area of the unit (n-1)-hemisphere, gives the probability that an orthogonal projection of S onto a random hyperplane is an (n-1)-simplex. Applications to more general polytopes are treated briefly, as is the related Perles-Shephard proof of the classical Gram-Euler relations.
A Bonnesen-type inequality is a sharp isoperimetric inequality that includes an error estimate in terms of inscribed and circumscribed regions. A kinematic technique is used to prove a Bonnesen-type inequality for the Euclidean sphere (having constant positive Gauss curvature) and the hyperbolic plane (having constant negative Gauss curvature). These generalized inequalities each converge to the classical Bonnesen-type inequality for the Euclidean plane as the curvature approaches zero.