In contrast to geometric similarity, perceptual similarity is manifold and much more difficult to handle. Relatively few work is known towards capturing the perceptual similarity. In this work we introduce the concept of local shape width and show that it can be applied to measure the perceptual significance of shape parts. Given such a measure, one can define the influence of the parts as a function of their significance. By doing so, we tend to base the shape matching on perceptually more meaningful parts and reduce the matching relevance of other parts. As an application, we propose a shape evolution approach, which synthesizes a series of new shapes from an input shape. They have the property that perceptually less significant parts smoothly vanish while other parts remain unchanged. This tool can be used in combination with any shape matching algorithm. A second application is proposed to perform non-uniform shape sampling. Experimental results will be given to show the practical usefulness of the local shape width concept.