Equivalent norms in ${{\mathbb{R}}}^{n}$ from thermodynamical laws

Abstract

In 1978, Landsberg proposed an elegant way of obtaining the inequality between arithmetic and geometric mean by using the first and second laws of thermodynamics. This result opened a debate on the logic legitimacy of this procedure to obtain some mathematical truths. Although this discussion can not be considered completed, the Landsberg approach has shown a great richness in obtaining many algebraic inequalities. In the present article we apply the Landsberg method to some properties of normed spaces trough a vector space of temperatures. In this way, the result that establishes the equivalence between all p-norms in the space n  and the minimal constant that guaranties this fact are readily found. Geometrical surfaces stemming from energy conservation are a consequence of this interpretation. In this manner, an application for thermal equilibrium of n reservoirs is suggested as an example of the contribution that the theory of norms may offer to physical problems when almost ideal heat baths are systems with a heat capacity of the form C aT m = with m  ¥. For this class of reservoirs the problem of obtaining the final temperature of two thermal baths with infinite or nearly infinite heat capacities in thermal contact is unambiguously solved. This might be the case in some systems which produce an incredibly large heat exchange due to small variations of temperature.

Extremal elements of a sublattice of the majorization lattice and approximate majorization

“…As it is well-known, all the ℓ p -norms are equivalent in nite dimensions (see, e.g., [43]). Indeed, for any p < q, one has [44] x q x p d…”