A Characteristic Intersection Property of Generalized Simplices

作者：Soltan Valeriu^*

来源：Journal of Convex Analysis, 2011, 18(2): 529-543.

• Homothety
• convex body
• intersection
• generalized simplex

Following Rockafellar (1970), a generalized n-simplex in R(n) is defined as the direct sum of an m-simplex and a simplicial (n - m)-cone, 0 <= m <= n. Fourneau (1977) showed that a line-free n-dimensional closed convex set K C R(n) is a generalized n-simplex if and only if all n-dimensional intersections K boolean AND (v + K), v is an element of R(n), are homothetic to K. We extend this characteristic property by proving that for a pair of line-free n-dimensional closed convex sets K(1) and K(2) in R(n) the following two conditions are equivalent: 1) all n-dimensional intersections K(1) boolean AND (v + K(2)), v E R(n), belong to a unique homothety class of convex sets, 2) K(1) and K(2) are generalized n-simplices whose n-dimensional intersections K(1) boolean AND (v + K(2)), v is an element of R(n), are homothetic to a unique generalized n-simplex.

• 出版日期2011