  The equivalence between non-extensive C. Tsallis entropy and the extensive
entropy introduced by Alfr\'ed R\'enyi is discussed. The R\'enyi entropy is
studied from the perspective of the geometry of the Lebesgue and generalised,
exotic Lebesgue spaces. A duality principle is established. The R\'enyi entropy
for the L\'evy distribution, in the domain when the nunerical methods fails, is
approximated by asymptotic expansion for the large values of the R\'enyi
parameter.
