A sharp Rogers and Shephard inequality for the 𝑝-difference body of planar convex bodies

Bianchini, Chiara; Colesanti, Andrea

doi:10.1090/S0002-9939-08-09209-5

A sharp Rogers and Shephard inequality for the $p$-difference body of planar convex bodies
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by Chiara Bianchini and Andrea Colesanti

Proc. Amer. Math. Soc. 136 (2008), 2575-2582

DOI: https://doi.org/10.1090/S0002-9939-08-09209-5

Published electronically: March 10, 2008

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Abstract:

We prove a sharp Rogers and Shephard type inequality for the $p$-difference body of a convex body in the two-dimensional case, for every $p\ge 1$.

References

Stefano Campi, Andrea Colesanti, and Paolo Gronchi, A note on Sylvester’s problem for random polytopes in a convex body, Rend. Istit. Mat. Univ. Trieste 31 (1999), no. 1-2, 79–94. MR 1763244
S. Campi and P. Gronchi, The $L^p$-Busemann-Petty centroid inequality, Adv. Math. 167 (2002), no. 1, 128–141. MR 1901248, DOI 10.1006/aima.2001.2036
Stefano Campi and Paolo Gronchi, On the reverse $L^p$-Busemann-Petty centroid inequality, Mathematika 49 (2002), no. 1-2, 1–11 (2004). MR 2059037, DOI 10.1112/S0025579300016004
Stefano Campi and Paolo Gronchi, On volume product inequalities for convex sets, Proc. Amer. Math. Soc. 134 (2006), no. 8, 2393–2402. MR 2213713, DOI 10.1090/S0002-9939-06-08241-4
Stefano Campi and Paolo Gronchi, Volume inequalities for $L_p$-zonotopes, Mathematika 53 (2006), no. 1, 71–80 (2007). MR 2304053, DOI 10.1112/S0025579300000036
Wm. J. Firey, $p$-means of convex bodies, Math. Scand. 10 (1962), 17–24. MR 141003, DOI 10.7146/math.scand.a-10510
Changqing Hu, Xi-Nan Ma, and Chunli Shen, On the Christoffel-Minkowski problem of Firey’s $p$-sum, Calc. Var. Partial Differential Equations 21 (2004), no. 2, 137–155. MR 2085300, DOI 10.1007/s00526-003-0250-9
Fritz John, Extremum problems with inequalities as subsidiary conditions, Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, Interscience Publishers, Inc., New York, 1948, pp. 187–204. MR 0030135
Erwin Lutwak, The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom. 38 (1993), no. 1, 131–150. MR 1231704
Erwin Lutwak, The Brunn-Minkowski-Firey theory. II. Affine and geominimal surface areas, Adv. Math. 118 (1996), no. 2, 244–294. MR 1378681, DOI 10.1006/aima.1996.0022
Erwin Lutwak, Deane Yang, and Gaoyong Zhang, $L_p$ affine isoperimetric inequalities, J. Differential Geom. 56 (2000), no. 1, 111–132. MR 1863023
Erwin Lutwak, Deane Yang, and Gaoyong Zhang, Sharp affine $L_p$ Sobolev inequalities, J. Differential Geom. 62 (2002), no. 1, 17–38. MR 1987375
Erwin Lutwak, Deane Yang, and Gaoyong Zhang, On the $L_p$-Minkowski problem, Trans. Amer. Math. Soc. 356 (2004), no. 11, 4359–4370. MR 2067123, DOI 10.1090/S0002-9947-03-03403-2
Mathieu Meyer and Shlomo Reisner, Shadow systems and volumes of polar convex bodies, Mathematika 53 (2006), no. 1, 129–148 (2007). MR 2304056, DOI 10.1112/S0025579300000061
C. A. Rogers and G. C. Shephard, The difference body of a convex body, Arch. Math. (Basel) 8 (1957), 220–233. MR 92172, DOI 10.1007/BF01899997
C. A. Rogers and G. C. Shephard, Convex bodies associated with a given convex body, J. London Math. Soc. 33 (1958), 270–281. MR 101508, DOI 10.1112/jlms/s1-33.3.270
C. A. Rogers and G. C. Shephard, Some extremal problems for convex bodies, Mathematika 5 (1958), 93–102. MR 104203, DOI 10.1112/S0025579300001418
Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. MR 1216521, DOI 10.1017/CBO9780511526282
G. C. Shephard, Shadow systems of convex sets, Israel J. Math. 2 (1964), 229–236. MR 179686, DOI 10.1007/BF02759738