Geodesic
Novel view on classical convexity theory
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 16 (2020) no. 3, pp. 291-311 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $B_{x}\subseteq\mathbb{R}^{n}$ denote the Euclidean ball with diameter $[0,x]$, i.e., with center at $\frac{x}{2}$ and radius $\frac{\left|x\right|}{2}$. We call such a ball a petal. A flower $F$ is any union of petals, i.e., $F=\bigcup_{x\in A}B_{x}$ for any set $A\subseteq\mathbb{R}^{n}$. We showed earlier in [9] that the family of all flowers $\mathcal{F}$ is in 1-1 correspondence with $\mathcal{K}_{0}$ – the family of all convex bodies containing $0$. Actually, there are two essentially different such correspondences. We demonstrate a number of different non-linear constructions on $\mathcal{F}$ and $\mathcal{K}_{0}$. Towards this goal we further develop the theory of flowers. 
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Vitali Milman; Liran Rotem. Novel view on classical convexity theory. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 16 (2020) no. 3, pp. 291-311. http://geodesic.mathdoc.fr/item/JMAG_2020_16_3_a5/

[1] S. Artstein-Avidan, A. Giannopoulos, V. Milman, Asymptotic Geometric Analysis, v. I, Mathematical Surveys and Monographs, 202, Amer. Math. Soc., Providence, RI, 2015 | DOI | MR | Zbl

[2] S. Artstein-Avidan, V. Milman, “The concept of duality for measure projections of convex bodies”, J. Funct. Anal., 254 (2008), 2648–2666 | DOI | MR | Zbl

[3] T. Bonnesen, W. Fenchel, Theory of Convex Bodies, BCS Associates, M.–Idaho, 1987 | MR | Zbl

[4] K.J. B{ö}r{ö}czky, E. Lutwak, D. Yang, G. Zhang, “The log-Brunn-Minkowski inequality”, Adv. Math., 231 (2012), 1974–1997 | DOI | MR

[5] J. Bourgain, J. Lindenstrauss, V. Milman, “Minkowski sums and symmetrizations”, Geometric Aspects of Functional Analysis, Israel Seminar 1986–1987, Lecture Notes in Mathematics, 1317, eds. J. Lindenstrauss, V. Milman, Springer, Berlin–Heidelberg, 1988, 44–66 | DOI | MR

[6] J. Bourgain, J. Lindenstrauss, V. Milman, “Approximation of zonoids by zonotopes”, Acta Math., 162 (1989), 73–141 | DOI | MR | Zbl

[7] T. Figiel, J. Lindenstrauss, V. Milman, “The dimension of almost spherical sections of convex bodies”, Acta Math., 139 (1977), 53–94 | DOI | MR | Zbl

[8] B.S. Kashin, “Diameters of some finite-dimensional sets and classes of smooth functions”, Mathematics of the USSR-Izvestiya, 11:2 (1977), 317–333 | DOI | MR | Zbl

[9] E. Milman, V. Milman, L. Rotem, “Reciprocals and flowers in convexity”, Geometric Aspects of Functional Analysis, Israel Seminar 2017–2019, v. II, Lecture Notes in Mathematics, 2266, eds. B. Klartag, E. Milman, Springer, Cham, 2020, 199–227 | DOI | MR | Zbl

[10] V. Milman, “New proof of the theorem of A. Dvoretzky on intersections of convex bodies”, Funct. Anal. Appl., 5 (1971), 288–295 | DOI | MR

[11] V. Milman, L. Rotem, ““Irrational” constructions in convex geometry”, Algebra i Analiz, 29 (2017), 222–236 | MR

[12] V. Milman, L. Rotem, “Powers and logarithms of convex bodies”, C. R. Math. Acad. Sci. Paris, 355:9 (2017), 981–986 | DOI | MR | Zbl

[13] V. Milman, L. Rotem, Weighted Geometric Means of Convex Bodies, Contemporary Mathematics, Amer. Math. Soc., Providence, RI, 2019 | MR | Zbl

[14] C. Saroglou, “More on logarithmic sums of convex bodies”, Mathematika, 62 (2016), 818–841 | DOI | MR | Zbl

[15] M. Schmuckenschl{ä}ger, “On the dependence on $\epsilon$ in a theorem of J. Bourgain, J. Lindenstrauss, V.D. Milman”, Geometric Aspects of Functional Analysis, Israel Seminar 1989–1990, Lecture Notes in Mathematics, 1469, eds. J. Lindenstrauss, V. Milman, Springer, Berlin–Heidelberg, 1991, 166–173 | DOI | MR

[16] B. Slomka, “On duality and endomorphisms of lattices of closed convex sets”, Adv. Geom., 11:2 (2011), 225–239 | DOI | MR | Zbl