TY - CHAP

T1 - Reciprocals and Flowers in Convexity

AU - Milman, Emanuel

AU - Milman, Vitali

AU - Rotem, Liran

N1 - Publisher Copyright:
© 2020, Springer Nature Switzerland AG.

PY - 2020

Y1 - 2020

N2 - 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 λi’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.

AB - 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 λi’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.

UR - http://www.scopus.com/inward/record.url?scp=85088515514&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-46762-3_9

DO - 10.1007/978-3-030-46762-3_9

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AN - SCOPUS:85088515514

T3 - Lecture Notes in Mathematics

SP - 199

EP - 227

BT - Lecture Notes in Mathematics

PB - Springer

ER -