Let [Formula: see text] be a closed Riemannian manifold and [Formula: see text] be the volume spectrum of [Formula: see text]. We will show that [Formula: see text] for all [Formula: see text], where [Formula: see text] and [Formula: see text] is the one-parameter Almgren–Pitts width of [Formula: see text]. We will also prove the analogous inequality for the [Formula: see text]-phase-transition spectrum [Formula: see text] using the Allen–Cahn approach.
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Akashdeep Dey (2024) studied this question.
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