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Isoperimetric ratio

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267: 240: 173: 142: 85:; any other curve has a larger value. Thus, the isoperimetric ratio can be used to measure how far from circular a shape is. 189: 289: 194: 74: 63: 59: 89: 67: 25: 138: 235:, Mathematical surveys and monographs, vol. 110, American Mathematical Society, p. 157, 263: 257: 236: 230: 169: 163: 17: 203: 134: 215: 211: 29: 283: 122: 93: 96:
so that, in the limit as the curve shrinks to a point, the ratio becomes 4
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Discrete Calculus: Applied Analysis on Graphs for Computational Science
207: 130: 82: 47: 133:(the measure of its interior). Other related quantities include the 55: 165:
Geometry Revealed: A Jacob's Ladder to Modern Higher Geometry
192:(1984), "Curve shortening makes convex curves circular", 107:, the isoperimetric ratio can similarly be defined as 77:, the isoperimetric ratio has its minimum value, 4 92:decreases the isoperimetric ratio of any smooth 125:of the body (the measure of its boundary) and 8: 103:For higher-dimensional bodies of dimension 256:Grady, Leo J.; Polimeni, Jonathan (2010), 154: 168:, Springer-Verlag, pp. 295–296, 7: 229:Chow, Bennett; Knopf, Dan (2004), 14: 262:, Springer-Verlag, p. 275, 232:The Ricci Flow: An Introduction 141:and the (differently defined) 1: 143:Cheeger constant of a graph 306: 68:similarity transformations 195:Inventiones Mathematicae 75:isoperimetric inequality 162:Berger, Marcel (2010), 60:dimensionless quantity 90:curve-shortening flow 139:Riemannian manifold 26:simple closed curve 22:isoperimetric ratio 208:10.1007/BF01388602 290:Analytic geometry 73:According to the 50:of the curve and 18:analytic geometry 297: 274: 272: 253: 247: 245: 226: 220: 218: 186: 180: 178: 159: 135:Cheeger constant 116: 99: 80: 53: 45: 41: 305: 304: 300: 299: 298: 296: 295: 294: 280: 279: 278: 277: 270: 255: 254: 250: 243: 228: 227: 223: 188: 187: 183: 176: 161: 160: 156: 151: 108: 97: 78: 51: 43: 33: 30:Euclidean plane 12: 11: 5: 303: 301: 293: 292: 282: 281: 276: 275: 268: 248: 241: 221: 202:(2): 357–364, 181: 174: 153: 152: 150: 147: 70:of the curve. 13: 10: 9: 6: 4: 3: 2: 302: 291: 288: 287: 285: 271: 269:9781849962902 265: 261: 260: 252: 249: 244: 242:9780821835159 238: 234: 233: 225: 222: 217: 213: 209: 205: 201: 197: 196: 191: 185: 182: 177: 175:9783540709978 171: 167: 166: 158: 155: 148: 146: 144: 140: 136: 132: 128: 124: 120: 115: 111: 106: 101: 95: 91: 86: 84: 76: 71: 69: 65: 61: 57: 49: 40: 36: 32:is the ratio 31: 27: 23: 19: 258: 251: 231: 224: 199: 193: 184: 164: 157: 126: 123:surface area 118: 113: 109: 104: 102: 94:convex curve 87: 72: 38: 34: 21: 15: 190:Gage, M. E. 149:References 58:. It is a 64:invariant 284:Category 81:, for a 62:that is 42:, where 216:0742856 129:is its 121:is the 54:is its 46:is the 28:in the 266:  239:  214:  172:  131:volume 117:where 83:circle 66:under 48:length 20:, the 137:of a 24:of a 264:ISBN 237:ISBN 170:ISBN 88:The 56:area 204:doi 16:In 286:: 212:MR 210:, 200:76 198:, 145:. 100:. 273:. 246:. 219:. 206:: 179:. 127:V 119:B 114:V 112:/ 110:B 105:d 98:π 79:π 52:A 44:L 39:A 37:/ 35:L

Index

analytic geometry
simple closed curve
Euclidean plane
length
area
dimensionless quantity
invariant
similarity transformations
isoperimetric inequality
circle
curve-shortening flow
convex curve
surface area
volume
Cheeger constant
Riemannian manifold
Cheeger constant of a graph
Geometry Revealed: A Jacob's Ladder to Modern Higher Geometry
ISBN
9783540709978
Gage, M. E.
Inventiones Mathematicae
doi
10.1007/BF01388602
MR
0742856
The Ricci Flow: An Introduction
ISBN
9780821835159
Discrete Calculus: Applied Analysis on Graphs for Computational Science

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