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The Lp-Busemann–Petty Centroid Inequality

✍ Scribed by S. Campi; P. Gronchi

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
137 KB
Volume
167
Category
Article
ISSN
0001-8708

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✦ Synopsis

The ratio between the volume of the p-centroid body of a convex body K in R n and the volume of K attains its minimum value if and only if K is an origin symmetric ellipsoid. This result, the L p -Busemann-Petty centroid inequality, was recently proved by Lutwak, Yang, and Zhang. In this paper we show that all the intrinsic volumes of the p-centroid body of K are convex functions of a time-like parameter when K is moved by shifting all the chords parallel to a fixed direction. The L p -Busemann-Petty centroid inequality is a consequence of this general fact.

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Some Lp Inequalities for the Polar Deriv
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✍ N.K Govil; Griffith Nyuydinkong; Berhanu Tameru 📂 Article 📅 2001 🏛 Elsevier Science 🌐 English ⚖ 95 KB

Let p n z be a polynomial of degree n and D α p n z its polar derivative. It has been proved that if p n z has no zeros in z < 1, then for δ ≥ 1 and α ≥ 1, 2π 0 D α p n e iθ δ dθ 1/δ ≤ n α + 1 F δ 2π 0 p n e iθ δ dθ 1/δ where F δ = 2π/ 2π 0 1 + e iθ δ dθ 1/δ . We also obtain analogous inequalities