A double-sided, indexable turning insert having a polygonal basic shape, including at least three corners, and an upper side and an under side, in which first and second opposite, plane support surfaces are included, The plane support surfaces are delimited inside outer contour lines, and parallel to each other and to a neutral plane (NP) situated halfway between the plane support surfaces. A plurality of cutting edges are along the upper side and the under side, each one of which includes a nose edge and two main edges converging toward the nose edge, all nose edges along the individual upper or under side being situated in a common reference plane (RP) that is depressed in relation to the support surface. A circumferential chip surface is adjacent to a peripheral clearance surface, the individual main edge being formed between the chip surface and a part of the clearance surface extending between two corners, and the individual nose edge being between the chip surface and a convex part of the clearance surface. A flank surface, which slopes toward the chip surface, extends from the outer contour line of the support surface. The individual main edge, as viewed in side elevation, declines from a highest point adjacent to the nose edge toward a lowest point between two corners, a level difference between the support surface and the lowest point of the main edge amounts to at most 0.5 mm, and the rake angle (α) of the individual cutting edge is acute along the entire extension of the cutting edge.