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.
Highlights
One of the key notions in convex analysis is the volume product M K K K o where K is a convex compact set of the Euclidean Rn, .,.with non-empty interior andK o = y Rn : x, y 1, x K is the so called polar set of K For this product, one of the long standing conjectures stated by Mahler claims that 4n n! MK for origin symmetric bodiesK
K o = y Rn : x, y 1, x K is the so called polar set of K For this product, one of the long standing conjectures stated by Mahler claims that for origin symmetric bodies
This conjecture has been confirmed in many special cases of K
Summary
Dimitris Karayannakis Department of Science, Division of Mathematics, TEI of Crete, Heraklion, Greece. -ball in Rn and q the Hölder conjugate exponent of function free of its gamma symbolism. This representausing 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. This representausing 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
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