= Solution
Choose a uniform lower bound $\kappa\in\mathbb R$ for the <Hessian matrix> of the <confinement potential>, so $D^2V(x)\geq\kappa I$ for every $x$. A sufficient condition is
$$
\boxed{\kappa+\lambda>0,\qquad\widetilde\lambda=\kappa+\lambda.}
$$
Indeed, <convexity> of $V(x)-\kappa|x|^2/2$ gives the <confinement energy> inequality from part (b), now with $\kappa$ possibly negative. Adding it to the equal-mean <interaction energy> inequality from part (d) gives
$$
\boxed{\mathcal E[\mu_t]\leq(1-t)\mathcal E[\mu]+t\mathcal E[\nu]-\frac{\kappa+\lambda}{2}t(1-t)d_2^2(\mu,\nu).}
$$
This proves positive <displacement convexity> on every class with fixed <barycenter>. In particular, if $D^2V\geq-KI$ with $K\geq0$, the sufficient condition is $\lambda>K$ and one may take $\widetilde\lambda=\lambda-K$. Wherever classical second derivatives exist, the equivalent differential calculation is $\mathcal E''[\mu_t]\geq(\kappa+\lambda)D^2$; the integrated proof also covers the printed growth assumptions.
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