Several geometric properties equivalent to non-reflexivity for a Banach space were studied by R.C. James in "Some self-dual properties of normed linear spaces". Ann. of Math. Studies 69 (1972), 159-175. See also Section 10 of D. van Dulst's book "Reflexive and superreflexive Banach spaces". Math. Centre Tracts 102. Amsterdam 1978. One of them:

A Banach space $X$ is non-reflexive if and only if there exists $\varepsilon>0$ and a sequence $(x_n)$ in the unit ball of $X$ such that for each $k\in N$, $$ dist(co\{x_1,\ldots,x_k\}, co\{x_{k+1},\ldots\})\geq \varepsilon). $$

Several geometric properties equivalent to non-reflexivity for a Banach space were studied by R.C. James in "Some self-dual properties of normed linear spaces". Ann. of Math. Studies 69 (1972), 159-175. See also Section 10 of D. van Dulst's book "Reflexive and superreflexive Banach spaces". Math. Centre Tracts 102. Amsterdam 1978. One of them:

A Banach space $X$ is non-reflexive if and only if there exist $\varepsilon>0$ and a sequence $(x_n)$ in the unit ball of $X$ such that for each $k\in N$, $$ dist(co\{x_1,\ldots,x_k\}, co\{x_{k+1},\ldots\})\geq \varepsilon. $$