We compare the volumes of projections of convex bodies and the volumes of the projections of their sections, and, dually, those of sections of convex bodies and of sections of their circumscribed cylinders. For L ⊂ R a convex body, we take n random segments in L and consider their ‘Minkowski average’ D. For fixed n, the pth moments of V (D) (1 ≤ p < ∞) are minimized, for V (L) fixed, by the ell...

Let L be a real (Hausdorff) topological vector space. The space %[L] of nonempty compact subsets of L forms a (Hausdorff) topological semivector space with singleton origin when 3C[L] is given the uniform (equivalently, the finite) hyperspace topology determined by L. Then %[L] is locally compact iff L is so. Furthermore, 90.[Z,], the set of nonempty compact convex subsets of L, is the largest ...

By a result of Helton and McCullough [HM12], open bounded convex free semialgebraic sets are exactly open (matricial) solution sets D◦ L of a linear matrix inequality (LMI) L(X) 0. This paper gives a precise algebraic certificate for a polynomial being nonnegative on a convex semialgebraic set intersect a variety, a so-called “Perfect” Positivstellensatz. For example, given a generic convex fre...

This paper will define the random walk on an integer lattice and will approximate the probability that the random walk is at a certain point after a certain number of steps by using a modified version of the Central Limit Theorem. To accomplish this, we will define the characteristic function of the random walk, find the Taylor expansion of this function, and bound the difference between this f...

We study the convex set L n deened by L n := fX j X = (x ij) positive semideenite n n matrix; x ii = 1 for all ig: We describe several geometric properties of L n. In particular, we show that L n has 2 n?1 vertices, which are its rank one matrices, corresponding to all bipartitions of the set f1; 2; : : : ; ng. Our main motivation for investigating the convex set L n comes from com-binatorial o...

1. The concept of arcwise convex set is a natural generalization of the notion of L set. The latter was studied by Horn and Valentine [2]1. It is especially interesting to observe how the theorem about the complement of an arcwise convex continuum sheds light on the corresponding theorem for L sets. In order to make this precise, the following definitions are used. Definition 1. An arc C is sai...

We give a description of an affine mapping T involving contact pairs of two general convex bodies K and L, when T (K) is in a position of maximal volume in L. This extends the classical John’s theorem of 1948, and is applied to the solution of a problem of Grünbaum; namely, any two convex bodies K and L in Rn have non-degenerate affine images K ′ and L′ such that K ′ ⊂ L′ ⊂ −nK ′. As a corollar...

A constructive algorithm to implement convex recursive deletion regions via two-layer perceptrons has been presented in a recent study. In the algorithm, the absolute values of the weights become larger and larger when the number of nested layers of a convex recursive deletion region increases. In addition, the absolute values of the weights are determined according to the complexity of the str...

Properties of several sorts of lattices of convex subsets of Rn are examined. The lattice of convex sets containing the origin turns out, for n > 1, to satisfy a set of identities strictly between those of the lattice of all convex subsets of Rn and the lattice of all convex subsets of R. The lattices of arbitrary, of open bounded, and of compact convex sets in Rn all satisfy the same identitie...