Convextaconvexa

Convextaconvexa is a concept in convex geometry describing objects that exhibit a bidirectional form of convexity under a specified duality. The term is used to capture the idea that a shape or set remains convex when viewed through two complementary perspectives, typically a primal and a dual description.

Definition: A convextaconvexa object is a pair (S, D) where S is a nonempty compact subset of a finite-dimensional real vector space and D is a duality operator on the family of convex bodies with the properties that D maps convex sets to convex sets and D is involutive (D(D(S)) = S). If, in addition, S equals the dual hull of D(S) in the sense of a prescribed duality, the object is said to be convextaconvexa.

Properties: Convextaconvexa objects are stable under affine transformations and under intersection with affine half-spaces, provided the duality is preserved. The class includes many familiar convex bodies as special cases (for example, when D is the identity, every convex body is convextaconvexa). If D corresponds to polarity with respect to a fixed gauge body K, convextaconvexa bodies often relate to self-polar or dual-pair configurations.

Examples: The unit disk in the Euclidean plane is convextaconvexa for the standard polarity with respect to the unit circle, and many symmetric polygons form convextaconvexa pairs with their duals.

Applications: The framework is used in educational contexts to illustrate dual convexity, in optimization theory to study bidirectional feasible regions, and in computational geometry for dual data representations.

History: The term convextaconvexa appears in contemporary discussions as a descriptive label for dual convexity, with varying formalizations across sources.