We investigate the Peak-Power Limited (PPL) Additive White Gaussian Noise (AWGN) channels in which the signal is band-limited, and its instantaneous power cannot exceed the power P. This model is relevant to many communication systems; however, its capacity is still unknown. We use a new geometry-based approach which evaluates the maximal entropy of the transmitted signal by assessing the volume of the body, in the space of Nyquist-rate samples, comprising all the points the transmitted signal can reach. This leads to lower bounds on capacity which are tight at high Signal to Noise Ratios (SNR). We found lower bounds on capacity, expressed as power efficiency, higher than the known ones by a factor of 3.35 and 8.6 in the low pass and the band pass cases respectively. The gap to the upper bounds is reduced to a power ratio of 1.5. The new bounds are numerically evaluated for FDMA-style signals with limited duration and also derived in the general case as a conjecture. The penalty in power efficiency due to the peak power constraint is roughly 6 dB at high SNR. Further research is needed to develop effective modulation and coding for this channel.