%0 Journal Article
%T A Volume Product Representation and Its Ramifications in &lt;i&gt;l&lt;sup&gt;n&lt;/sup&gt;&lt;sub&gt;p&lt;/sub&gt;&lt;/i&gt;, 1≤&lt;i&gt;p&lt;/i&gt;≤∞
%A Dimitris Karayannakis
%J Advances in Pure Mathematics
%P 264-266
%@ 2160-0384
%D 2011
%I Scientific Research Publishing
%R 10.4236/apm.2011.15046
%X Let|&lt;i&gt;B&lt;sup&gt;n&lt;/sup&gt;&lt;sub&gt;p&lt;/sub&gt;&lt;/i&gt;|,1＜&lt;i&gt;p&lt;/i&gt;＜∞ ,  be the volume of the unit  &lt;i&gt;p&lt;/i&gt;-ball in  &lt;i&gt;R&lt;sup&gt;n&lt;/sup&gt;&lt;/i&gt; and &lt;i&gt;q&lt;/i&gt;  the H&#246;lder conjugate exponent of  &lt;i&gt;p&lt;/i&gt;. We represent the volume product |&lt;i&gt;B&lt;sup&gt;n&lt;/sup&gt;&lt;sub&gt;p&lt;/sub&gt;&lt;/i&gt;| |&lt;i&gt;B&lt;sup&gt;n&lt;/sup&gt;&lt;sub&gt;a&lt;/sub&gt;&lt;/i&gt;|   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 &lt;i&gt;R&lt;sup&gt;n&lt;/sup&gt;&lt;/i&gt; . These bounds in the general case play a central role in convex geometric analysis.
%K P-Ball
%K Volume
%K Gamma Function
%K Infinite Product
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=7254