The owners of lots or portions of lots fronting on any public street or place when that street or place has been improved by the construction of sidewalks or curbs for a total frontage of more than fifty percent (50%) on one side of such street or place in any block, or where a petition signed by the owners of more than sixty percent (60%) of the front footage of the block has been filed with the city clerk requesting the installation of such improvements, or where a petition signed by the owners of more than sixty percent (60%) of the front footage of any part of an unimproved portion or portions of a block has been filed with the city clerk requesting the installation of such improvements in front of said part, or whenever the legislative body of the city upon its own motion orders the installation of such improvements in front of said part, shall have the duty of constructing or causing the construction of sidewalks or curbs in front of their properties upon notice so to do by the superintendent of streets.