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We give a new scale of completeness conditions for exponential systems in twotypes of functional spaces on subsets of the complex plane. The first is theBanach spaces of functions that are continuous on a compact and simultaneouslyholomorphic in the interior of this compact, if this interior is nonempty, witha uniform norm. The second is the spaces of holomorphic functions on a boundedopen set with a topology of uniform convergence on compacts. These conditionsare formulated in terms of the perimeter of the convex hull of the domain ofdetermining of functions from space and new characteristics of distribution ofexponents of exponential system.

Completeness of exponential systems and the perimeter of the convex hull