A Volume Product Representation and Its Ramifications in lnp, 1≤p≤∞
Author(s)
Dimitris Karayannakis
ABSTRACT
Let|Bnp|,1＜p＜∞ , be the volume of the unit p-ball in Rn and q the Hölder conjugate exponent of p. We represent the volume product |Bnp| |Bna| as a function free of its gamma symbolism. This representation will allows us in this particular case to confirm, using basic classical analysis tools, two conjectured and partially proved lower and upper bounds for the volume product of centrally symmetric convex bodies of the Euclidean Rn . These bounds in the general case play a central role in convex geometric analysis.
Let|Bnp|,1＜p＜∞ , be the volume of the unit p-ball in Rn and q the Hölder conjugate exponent of p. We represent the volume product |Bnp| |Bna| as a function free of its gamma symbolism. This representation will allows us in this particular case to confirm, using basic classical analysis tools, two conjectured and partially proved lower and upper bounds for the volume product of centrally symmetric convex bodies of the Euclidean Rn . These bounds in the general case play a central role in convex geometric analysis.
Cite this paper
nullD. Karayannakis, "A Volume Product Representation and Its Ramifications in lnp, 1≤p≤∞," Advances in Pure Mathematics, Vol. 1 No. 5, 2011, pp. 264-266. doi: 10.4236/apm.2011.15046.
nullD. Karayannakis, "A Volume Product Representation and Its Ramifications in lnp, 1≤p≤∞," Advances in Pure Mathematics, Vol. 1 No. 5, 2011, pp. 264-266. doi: 10.4236/apm.2011.15046.
References
[1] G. Andrews, R. Askey and R. Roy, “Special Functions,” Cambridge University Press, Cambridge, 1999. [2] I. Gradshteyn and I. Ryzhik, “Tables of Integrals, Series and Products,” Academic Press, Waltham, 1980. [3] D. Karayannakis, “An algorithm for the Evaluation of the Gamma function and Ramifications. Part I,” International Journal of Mathematics, Game Theory and Algebra, Vol. 19, No. 4, 2010. [4] L. E. Lutwak, “Selected Affine Isoperimetric Inequalities, Handbook of Convex Geometry,” North-Holland Publishing Co., Amsterdam, 1993.
[1] G. Andrews, R. Askey and R. Roy, “Special Functions,” Cambridge University Press, Cambridge, 1999. [2] I. Gradshteyn and I. Ryzhik, “Tables of Integrals, Series and Products,” Academic Press, Waltham, 1980. [3] D. Karayannakis, “An algorithm for the Evaluation of the Gamma function and Ramifications. Part I,” International Journal of Mathematics, Game Theory and Algebra, Vol. 19, No. 4, 2010. [4] L. E. Lutwak, “Selected Affine Isoperimetric Inequalities, Handbook of Convex Geometry,” North-Holland Publishing Co., Amsterdam, 1993.