[CloudFront] Check specifically for undefined and determining of the defaultRootObject prop is set or not

packages/@aws-cdk/aws-cloudfront/lib/web_distribution.ts

@@ -459,7 +459,7 @@ export class CloudFrontWebDistribution extends cdk.Construct {
     const distributionConfig: cloudformation.DistributionResource.DistributionConfigProperty = {
       comment: props.comment,
       enabled: true,
-      defaultRootObject: props.defaultRootObject || "index.html",
+      defaultRootObject: props.defaultRootObject !== undefined ? props.defaultRootObject : "index.html",
       httpVersion: props.httpVersion || HttpVersion.HTTP2,
       priceClass: props.priceClass || PriceClass.PriceClass100,
       ipv6Enabled: props.enableIpV6 || true,

packages/@aws-cdk/aws-cloudfront/test/integ.cloudfront-empty-root.expected.json

@@ -0,0 +1,60 @@
+{
+  "Resources": {
+    "AnAmazingWebsiteProbablyCFDistribution47E3983B": {
+      "Type": "AWS::CloudFront::Distribution",
+      "Properties": {
+        "DistributionConfig": {
+          "CacheBehaviors": [],
+          "DefaultCacheBehavior": {
+            "AllowedMethods": [
+              "GET",
+              "HEAD"
+            ],
+            "CachedMethods": [
+              "GET",
+              "HEAD"
+            ],
+            "ForwardedValues": {
+              "Cookies": {
+                "Forward": "none"
+              },
+              "QueryString": false
+            },
+            "TargetOriginId": "origin1",
+            "ViewerProtocolPolicy": "redirect-to-https"
+          },
+          "DefaultRootObject": "",
+          "Enabled": true,
+          "HttpVersion": "http2",
+          "IPV6Enabled": true,
+          "Origins": [
+            {
+              "CustomOriginConfig": {
+                "HTTPPort": 80,
+                "HTTPSPort": 443,
+                "OriginKeepaliveTimeout": 5,
+                "OriginProtocolPolicy": "https-only",
+                "OriginReadTimeout": 30,
+                "OriginSSLProtocols": [
+                  "TLSv1.2"
+                ]
+              },
+              "DomainName": "brelandm.a2z.com",
+              "Id": "origin1",
+              "OriginCustomHeaders": [
+                {
+                  "HeaderName": "X-Custom-Header",
+                  "HeaderValue": "somevalue"
+                }
+              ]
+            }
+          ],
+          "PriceClass": "PriceClass_100",
+          "ViewerCertificate": {
+            "CloudFrontDefaultCertificate": true
+          }
+        }
+      }
+    }
+  }
+}

packages/@aws-cdk/aws-cloudfront/test/integ.cloudfront-empty-root.ts

@@ -0,0 +1,28 @@
+
+import cdk = require('@aws-cdk/cdk');
+import cloudfront = require('../lib');
+
+const app = new cdk.App(process.argv);
+
+const stack = new cdk.Stack(app, 'aws-cdk-cloudfront-custom');
+
+new cloudfront.CloudFrontWebDistribution(stack, 'AnAmazingWebsiteProbably', {
+  originConfigs: [
+    {
+      originHeaders: {
+        "X-Custom-Header": "somevalue",
+      },
+      customOriginSource: {
+        domainName: "brelandm.a2z.com",
+      },
+      behaviors: [
+        {
+        isDefaultBehavior: true,
+        }
+      ],
+    }
+  ],
+  defaultRootObject: ''
+});
+
+process.stdout.write(app.run());