Plurisubharmonic geodesics and interpolating sets

Abstract

We apply a notion of geodesics of plurisubharmonic functions to interpolation of compact subsets of C n . Namely, two non-pluripolar, polynomially closed, compact subsets of C n are interpolated as level sets L t = {z : u t (z) = −1} for the geodesic u t between their relative extremal functions with respect to any ambient bounded domain. The sets L t are described in terms of certain holomorphic hulls. In the toric case, it is shown that the relative Monge-Ampère capacities of L t satisfy a dual Brunn-Minkowski inequality.

Interpolation of Weighted Extremal Functions

“…Here, we consider the general case of u c j = c j ω K j with c j > 0 and K j non-pluripolar, compact, polynomially convex subsets of a bounded hyperconvex domain Ω of C n . In this situation, the functions u c j (z) = −c j precisely on K j and are continuous on Ω, the geodesics u t converge to u j , uniformly on Ω, as t → j, and belong to C(Ω × [0, 1]), as in the non-weighted case dealt with in [18] and [8].…”

“…Classical complex interpolation of Banach spaces, due to Calderón [5] (see [3] and, for more recent developments, [7]) is based on constructing holomorphic hulls generated by certain families of holomorphic mappings. A slightly different approach proposed in [8] rests on plurisubharmonic geodesics. The notion has been originally considered, starting from 1987, for metrics on compact Kähler manifolds (see [10] and the bibliography therein), while its local counterpart for plurisubharmonic functions on bounded hyperconvex domains of C n was introduced more recently in [4] and [18], see also [1].…”

“…Moreover, if the sets K j are Reinhardt (toric), then u t = ω K t for some t ∈ (0, 1) only if K 0 = K 1 , so an equality in (3) is never possible unless the geodesic degenerates to a point. Furthermore, in the toric case, the capacities (with respect to the unit polydisk D n ) were proved in [8] to be not just convex functions of t, as is depicted in (3), but logarithmically convex:…”

“…Finally, we transfer the above results on the capacities of sets in C n to the realm of convex geometry, developing thus the Brunn-Minkowski theory for volumes of (co)convex sets in R n [8,12,19].…”

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An approach to interpolation of compact subsets of C n , including Brunn-Minkowski type inequalities for the capacities of the interpolating sets, was developed in [8] by means of plurisubharmonic geodesics between relative extremal functions of the given sets. Here we show that a much better control can be achieved by means of the geodesics between weighted relative extremal functions.

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