116 research outputs found
Operations between sets in geometry
An investigation is launched into the fundamental characteristics of
operations on and between sets, with a focus on compact convex sets and star
sets (compact sets star-shaped with respect to the origin) in -dimensional
Euclidean space . For example, it is proved that if , with three
trivial exceptions, an operation between origin-symmetric compact convex sets
is continuous in the Hausdorff metric, GL(n) covariant, and associative if and
only if it is addition for some . It is also
demonstrated that if , an operation * between compact convex sets is
continuous in the Hausdorff metric, GL(n) covariant, and has the identity
property (i.e., for all compact convex sets , where
denotes the origin) if and only if it is Minkowski addition. Some analogous
results for operations between star sets are obtained. An operation called
-addition is generalized and systematically studied for the first time.
Geometric-analytic formulas that characterize continuous and GL(n)-covariant
operations between compact convex sets in terms of -addition are
established. The term "polynomial volume" is introduced for the property of
operations * between compact convex or star sets that the volume of ,
, is a polynomial in the variables and . It is proved that if
, with three trivial exceptions, an operation between origin-symmetric
compact convex sets is continuous in the Hausdorff metric, GL(n) covariant,
associative, and has polynomial volume if and only if it is Minkowski addition
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