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In this paper we deal with the min-max invariant known as p p -width for the 3-dimensional real projective space. More precisely, we present an explicit and sharp p p -sweepout, for p = 1 p=1 , 2 2 , 3 3 , and compute the value of the p p -width for such values. Using Lusternik-Schnirelmann type argument we also verify the jump of the 5 5 -width and, using algebraic sets, we estimate the 9 9 -width.

In this paper, we deal with the first and second widths of the real projective space R P n \mathbb {RP}^{n} , for n n ranging from 4 4 to 7 7 , and for this we used some tools from the Almgren-Pitts min-max theory. In a recent paper, Ramirez-Luna computed the first width of the real projective spaces, and, at the same time, we obtained optimal sweepouts realizing the first and second widths of those spaces.

In this note, we explore the nature of Lens spaces to study the first width of those spaces, more precisely, we use the existence of a sharp sweep out associated to a Clifford torus to provide a simple and pretty application of the Willmore conjecture for the computation of the [Formula: see text]-width of Lens spaces.