A Volume Product Representation and Its Ramifications in <i>lⁿ_p</i>, 1≤<i>p</i>≤∞

Abstract

Let|<i>Bⁿ_p</i>|,1＜<i>p</i>＜∞ , be the volume of the unit <i>p</i>-ball in <i>Rⁿ</i> and <i>q</i> the Hölder conjugate exponent of <i>p</i>. We represent the volume product |<i>Bⁿ_p</i>| |<i>Bⁿ_a</i>| 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 <i>Rⁿ</i> . These bounds in the general case play a central role in convex geometric analysis.