Table 3

Shape functionals for simply connected compact sets. A, P, r, R, ω, d, denote the area, perimeter, radii of the inscribed and circumscribed circles, minimum and maximum Feret diameters [24], respectively.

Geometrical functionals Geometric inequalities Morphological functionals Extremal sets
r, R r ≤ R r/R C
ω, R ω ≤ 2R ω/2R C
A, R A ≤ πR2 A/πR2 C
d, R d ≤ 2R d/2R Y
r, d 2r ≤ d 2r/d C
ω, d ω ≤ d ω/d W
A, d 4A ≤ πd2 4A/πd2 C
R, d Z
r, P 2πr ≤ P 2πr/P C
ω, P πω ≤ P W
A, P 4πA ≤ P2 4πA/P2 C
d, P 2d ≤ P 2d/P L
R, P 4R ≤ P 4R/P L
r, A πr2 ≤ A πr2/A C
r, ω 2r ≤ ω 2r/ω X

Extremal sets are the sets for which an inequality becomes an equality.

C: the disks; W: the constant width compact convex sets; L: the line segments; X: many compact convex sets; Y: many simply connected compact sets; Z every compact convex set of diameter d containing an equilateral triangle of side-length d.

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