Sublinear Neural Networks Parametrize Convex Sets

A novel approach employs sublinear neural networks to characterize convex sets through the learning of positively homogeneous and convex functions. These networks serve to implicitly define both support and gauge functions associated with convex bodies. A proof is provided for a universal approximation theorem applicable to convex sets using this parametrization. Empirical findings related to shape optimization and inverse design tasks demonstrate the precise reconstruction of desired shapes.