We introduce a new class of distinguished regions based on detecting the most salient convex local arrangements of contours in the image. The regions are used in a similar way to the local interest points extracted from gray-level images, but they capture shape rather than texture. Local convexity is characterized by measuring the extent to which the detected image contours support circle or arc-like local structures at each position and scale in the image. Our saliency measure combines two cost functions defined on the tangential edges near the circle: a tangential-gradient energy term, and an entropy term that ensures local support from a wide range of angular positions around the circle. The detected regions are invariant to scale changes and rotations, and robust against clutter, occlusions and spurious edge detections. Experimental results show very good performance for both shape matching and recognition of object categories.