We consider the problem of secure state estimation in an adversarial environment with the presence of bounded noises. We assume the adversary has the knowledge of the healthy measurements and system parameters. To countervail the dangerous attacker, the problem is given as a min-max optimization, that is, the system operator seeks an estimator which minimizes the worst-case estimation error due to the manipulation by the attacker. On the proposed estimator, the estimation error is bounded at all times even if the system removing an arbitrary set of 2l sensors is not observable, where l is the number of the compromised sensors. To this end, taking the reach set of the system into account, we first show the feasible set of the state can be represented as a union of polytopes, and the optimal estimate is given as the Chebyshev center of the union. Then, for calculating the optimal state estimate, we provide a convex optimization problem that utilizes the vertices of the union. Additionally, the upper bound of the worst-case estimation error is derived theoretically, and we also show a rigorous analytical bound under a certain condition. The attacked sensor identification algorithm is further provided. A simple numerical example finally shows to illustrate the effectiveness of the proposed estimator.