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...

A new method for the design of fixed-order dynamic output-feedback Linear Parameter Varying (LPV) controllers for discrete-time LPV systems with bounded scheduling parameter variations is presented. Sufficient conditions for stability and induced l2-norm performance of an LPV system are given through a set of Linear Matrix Inequalities (LMIs) and exploited for design. Controller parameters appe...

Let V H ðC Þ be a variety and let Log denote the logarithmic moment map Log : ðC Þ ! R, ðz1; . . . ; znÞ 7! ðlogjz1j; . . . ; logjznjÞ. The amoeba A of the variety V H ðC Þ is its image under that map. Amoebas were introduced by Gelfand, Kapranov and Zelevinsky in [6]. They show that for varieties of codimension 1, the complement A of the amoeba A in R is a disjoint union of convex regions. If ...

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...

In this article, we are concerned by existence and uniqueness of a fixed point for the sum two operators A B, defined on closed convex subset an ordered Banach space, where order is induced normal minihedral cone. such structure, absolute value function generated provide ability to introduce new versions concepts lipschitzian expansive mappings. Therefore prove that if B contractive, then + has...