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The concept of an $i$-symmetrization is introduced, which provides a convenient framework for most of the familiar symmetrization processes on convex sets. Various properties of $i$-symmetrizations are introduced and the relations between them investigated. New expressions are provided for the Steiner and Minkowski symmetrals of a compact convex set which exhibit a dual relationship between them. Characterizations of Steiner, Minkowski and central symmetrization, in terms of natural properties that they enjoy, are given and examples are provided to show that none of the assumptions made can be dropped or significantly weakened. Other familiar symmetrizations, such as Schwarz symmetrization, are discussed and several new ones introduced.
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Formulas about the side lengths, diagonal lengths or radius of the circumcircle of a cyclic polygon in Euclidean geometry, hyperbolic geometry or spherical geometry can be unified.
Let $(X, d)$ be a compact metric space and let $mathcal{M}(X)$ denote the space of all finite signed Borel measures on $X$. Define $I colon mathcal{M}(X) to R$ by [I(mu) = int_X int_X d(x,y) dmu(x) dmu(y),] and set $M(X) = sup I(mu)$, where $mu$ ranges over the collection of signed measures in $mathcal{M}(X)$ of total mass 1. The metric space $(X, d)$ is quasihypermetric if for all $n in N$, all $alpha_1, ..., alpha_n in R$ satisfying $sum_{i=1}^n alpha_i = 0$ and all $x_1, ..., x_n in X$, one has $sum_{i,j=1}^n alpha_i alpha_j d(x_i, x_j) leq 0$. Without the quasihypermetric property $M(X)$ is infinite, while with the property a natural semi-inner product structure becomes available on $mathcal{M}_0(X)$, the subspace of $mathcal{M}(X)$ of all measures of total mass 0. This paper explores: operators and functionals which provide natural links between the metric structure of $(X, d)$, the semi-inner product space structure of $mathcal{M}_0(X)$ and the Banach space $C(X)$ of continuous real-valued functions on $X$; conditions equivalent to the quasihypermetric property; the topological properties of $mathcal{M}_0(X)$ with the topology induced by the semi-inner product, and especially the relation of this topology to the weak-$*$ topology and the measure-norm topology on $mathcal{M}_0(X)$; and the functional-analytic properties of $mathcal{M}_0(X)$ as a semi-inner product space, including the question of its completeness. A later paper [Peter Nickolas and Reinhard Wolf, Distance Geometry in Quasihypermetric Spaces. II] will apply the work of this paper to a detailed analysis of the constant $M(X)$.
Let $(X, d)$ be a compact metric space and let $mathcal{M}(X)$ denote the space of all finite signed Borel measures on $X$. Define $I colon mathcal{M}(X) to R$ by [ I(mu) = int_X int_X d(x,y) dmu(x) dmu(y), ] and set $M(X) = sup I(mu)$, where $mu$ ranges over the collection of signed measures in $mathcal{M}(X)$ of total mass 1. This paper, with an earlier and a subsequent paper [Peter Nickolas and Reinhard Wolf, Distance geometry in quasihypermetric spaces. I and III], investigates the geometric constant $M(X)$ and its relationship to the metric properties of $X$ and the functional-analytic properties of a certain subspace of $mathcal{M}(X)$ when equipped with a natural semi-inner product. Using the work of the earlier paper, this paper explores measures which attain the supremum defining $M(X)$, sequences of measures which approximate the supremum when the supremum is not attained and conditions implying or equivalent to the finiteness of $M(X)$.
Let $(X, d)$ be a compact metric space and let $mathcal{M}(X)$ denote the space of all finite signed Borel measures on $X$. Define $I colon mathcal{M}(X) to R$ by [ I(mu) = int_X int_X d(x,y) dmu(x) dmu(y), ] and set $M(X) = sup I(mu)$, where $mu$ ranges over the collection of signed measures in $mathcal{M}(X)$ of total mass 1. This paper, with two earlier papers [Peter Nickolas and Reinhard Wolf, Distance geometry in quasihypermetric spaces. I and II], investigates the geometric constant $M(X)$ and its relationship to the metric properties of $X$ and the functional-analytic properties of a certain subspace of $mathcal{M}(X)$ when equipped with a natural semi-inner product. Specifically, this paper explores links between the properties of $M(X)$ and metric embeddings of $X$, and the properties of $M(X)$ when $X$ is a finite metric space.
Given two $n_i$-dimensional Alexandrov spaces $X_i$ of curvature $ge 1$, the join of $X_1$ and $X_2$ is an $(n_1+n_2+1)$-dimensional Alexandrov space $X$ of curvature $ge 1$, which contains $X_i$ as convex subsets such that their points are $frac pi2$ apart. If a group acts isometrically on a join that preserves $X_i$, then the orbit space is called quotient of join. We show that an $n$-dimensional Alexandrov space $X$ with curvature $ge 1$ is isometric to a finite quotient of join, if $X$ contains two compact convex subsets $X_i$ without boundary such that $X_1$ and $X_2$ are at least $frac pi2$ apart and $dim(X_1)+dim(X_2)=n-1$.
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