Reciprocals and Flowers in Convexity

We study new classes of convex bodies and star bodies with unusual properties. First we define the class of reciprocal bodies, which may be viewed as convex bodies of the form “1∕K”. The map K↦K^′ sending a body to its reciprocal is a duality on the class of reciprocal bodies, and we study its properties. To connect this new map with the classic polarity we use another construction, associating to each convex body K a star body which we call its flower and denote by K^♣. The mapping K↦K^♣ is a bijection between the class K0n of convex bodies and the class ℱⁿ of flowers. Even though flowers are in general not convex, their study is very useful to the study of convex geometry. For example, we show that the polarity map ∘:K0n→K0n decomposes into two separate bijections: First our flower map ♣:K0n→ℱn, followed by a slight modification Φ of the spherical inversion which maps ℱⁿ back to K0n. Each of these maps has its own properties, which combine to create the various properties of the polarity map. We study the various relations between the four maps ′, ∘, ♣ and Φ and use these relations to derive some of their properties. For example, we show that a convex body K is a reciprocal body if and only if its flower K^♣ is convex. We show that the class ℱⁿ has a very rich structure, and is closed under many operations, including the Minkowski addition. This structure has corollaries for the other maps which we study. For example, we show that if K and T are reciprocal bodies so is their “harmonic sum” (K^∘ + T^∘)^∘. We also show that the volume|(∑iλiKi)♣| is a homogeneous polynomial in the λ_i’s, whose coefficients can be called “♣-type mixed volumes”. These mixed volumes satisfy natural geometric inequalities, such as an elliptic Alexandrov–Fenchel inequality. More geometric inequalities are also derived.