Beamforming Gain Maximization for Fluid Reconfigurable Intelligent Surface: A Minkowski Geometry Approach

This paper investigates beamforming-gain maximization for a fluid reconfigurable intelligent surface (FRIS)-assisted downlink system, where each active port applies a finite-resolution unit-modulus phase selected from a discrete codebook. The resulting design couples the multi-antenna base-station (BS) beamformer with combinatorial FRIS port selection and discrete phase assignment, leading to a highly nonconvex mixed discrete optimization. To address this challenge, we develop an alternating-optimization (AO) framework that alternates between a closed-form maximum-ratio-transmission (MRT) update at the BS and an {optimal} FRIS-configuration update. The key step of the proposed FRIS configuration is a Minkowski-geometry reformulation of the FRIS codebook superposition: by convexifying the feasible reflected-sum set and exploiting support-function identities, we convert the FRIS subproblem into a one-dimensional maximization over a directional parameter. For each direction, the optimal configuration is obtained constructively via per-port directional scoring, Top-$M_o$ port selection, and optimal codeword assignment. For the practically important regular $M_p$-gon phase-shifter codebook, we further derive closed-form score expressions and establish a piecewise-smooth structure of the resulting support function, which leads to a finite critical-angle search that provably identifies the global optimum without exhaustive angular sweeping. Simulation results demonstrate that the proposed framework consistently outperforms benchmarks, achieves near-optimal beamforming gains in exhaustive-search validations, accurately identifies the optimal direction via support-function maximization, and converges rapidly within a few AO iterations.