Abstract
We study the volume of unit balls B-p,q(n) of finite-dimensional Lorentz sequence spaces l(p,q)(n). We give an iterative formula for vol(B-p,q(n)) for the weak Lebesgue spaces with q = infinity and explicit formulas for q = 1 and q = infinity.
We derive asymptotic results for the nth root of vol(B-p,q(n)) and show that [vol(B-p,q(n)](1/n) asymptotic to(p,q) n(-1/p) for all 0 < p < infinity and 0 < q <= infinity. We study further the ratio between the volume of unit balls of weak Lebesgue spaces and the volume of unit balls of classical Lebesgue spaces.
We conclude with an application of the volume estimates and characterize the decay of the entropy numbers of the embedding of the weak Lebesgue space l(p),(n)(infinity) into l(p)(n). (C) 2020 Elsevier Inc. All rights reserved.