is the scalar product on R. The multidimensional transform can be interpreted as an encoding of the convex hull of the function's epigraph in terms of its supporting hyperplanes. This can be seen as consequence of the following two observations. On the one hand, the hyperplane tangent to the epigraph of f at some point ( x , f ( x ) ) ∈ ∈ U × × R has normal vector ( ∇ ∇ f ( x ) , − − 1 ) ∈ ∈ R n + 1 . On the other hand, any closed convex set C ∈ ∈ R m can be characterized via the set of its supporting hyperplanes by the equations x ⋅ ⋅ n = h C ( n ) , where h C ( n ) is the support function of C . But the definition of Legendre transform via the maximization matches precisely that of the support function, that is, f ∗ ∗ ( x ) = h epi ⁡ ⁡ ( f ) ( x , − − 1 ) . We thus conclude that the Legendre transform characterizes the epigraph in the sense that the tangent plane to the epigraph at any point ( x , f ( x ) ) is given explicitly by { z ∈ ∈ R n + 1 : z ⋅ ⋅ x = f ∗ ∗ ( x ) } .