$C^{1,\alpha }$ regularity of variational problems with a convexity constraint

In this paper, we establish the interior $C^{1,\alpha }$ regularity of minimizers of a class of functionals with a convexity constraint, which includes the principal-agent problems studied by Figalli-Kim-McCann ($\mathit{\text{J.}}\mathit{\text{Econom.}}\mathit{\text{Theory}}$ $\mathbf{\text{146}}$ (2011), no. 2, 454-478). The $C^{1,1}$ regularity was previously proved by Caffarelli-Lions in an unpublished note when the cost is quadratic, and recently extended to the case where the cost is uniformly convex with respect to a general preference function by McCann-Rankin-Zhang($\mathit{\text{arXiv:2303.04937v3}}$). Our main result does not require the uniform convexity assumption on the cost function. In particular, we show that the solutions to the principal-agent problems with $q$-power cost are $C^{1,\frac{1}{q-1}}$ when $q>2$ and $C^{1,1}$ when $1<q\le 2$. Examples can show that this regularity is optimal when $q\ge 2$.